Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 8 (2012), 027, 45 pages
Emergent Models for Gravity:
an Overview of Microscopic Models?
Lorenzo SINDONI
Max Planck Institute for Gravitational Physics, Albert Einstein Institute,
Am Mühlenberg 1, 14467 Golm, Germany
E-mail: sindoni@aei.mpg.de
Received October 02, 2011, in final form May 04, 2012; Published online May 12, 2012
http://dx.doi.org/10.3842/SIGMA.2012.027
Abstract. We give a critical overview of various attempts to describe gravity as an emergent phenomenon, starting from examples of condensed matter physics, to arrive to more
sophisticated pregeometric models. The common line of thought is to view the graviton
as a composite particle/collective mode. However, we will describe many different ways in
which this idea is realized in practice.
Key words: emergent gravity; quantum gravity
2010 Mathematics Subject Classification: 83C45; 83C50; 81Q20
Contents
1 Introduction
2
2 A warm up
2.1 Bjorken’s model for electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Induced gravity versus emergent gravity . . . . . . . . . . . . . . . . . . . . . . .
3
4
5
3 Emergent spacetimes: lessons from condensed matter
3.1 BEC systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Fermi point scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
8
11
4 The fate of Lorentz invariance
4.1 Multi-component systems: normal modes analysis . . . . . . . . . . . . . . . . .
4.2 From analogue models to Finsler geometry . . . . . . . . . . . . . . . . . . . . . .
12
12
14
5 Analogue nonrelativistic dynamics: the BEC strikes back
5.1 Single BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Lessons: Poisson equation, mass and the cosmological constant . . . . . . . . . .
5.3 Multi-BEC, masslessness and the equivalence principle . . . . . . . . . . . . . . .
16
17
19
20
6 Background Minkowski spacetime: the graviton as a
6.1 Comments on massless particles . . . . . . . . . . . . .
6.2 Gauge invariance from spontaneous Lorentz violation .
6.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . .
21
21
23
25
?
Goldstone
. . . . . . .
. . . . . . .
. . . . . . .
boson
. . . . . . . .
. . . . . . . .
. . . . . . . .
This paper is a contribution to the Special Issue “Loop Quantum Gravity and Cosmology”. The full collection
is available at http://www.emis.de/journals/SIGMA/LQGC.html
2
L. Sindoni
7 Towards pregeometry
7.1 Spin systems and quantum graphity . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Emerging diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
26
28
8 Background independent pregeometrical models
29
9 Lorentz invariance reloaded: emergent signature
33
10 Challenges and future directions
35
References
37
1
Introduction
Despite some compelling theories have been proposed to finally merge quantum theory with
gravitation [184, 191], it is fair to say that we are still away from a definitive answer to the
question “what is the microscopic structure of spacetime?”.
Interestingly enough, classical general relativity does possess in itself some peculiar signatures
that link it to thermodynamics: from the most celebrated results about black hole thermodynamics [37, 40] to the more recent results establishing a connection between thermodynamics of
local horizons and general relativity (as well as other theories of gravity) [67, 68, 92, 138, 176,
177, 220], the AdS/CFT correspondence [4] and the gravity/fluid connection (see, e.g., [136]).
These results show that it is not unconceivable that general relativity is ultimately a form of hydrodynamics for the microscopic constituents whose collective, large scale behavior is described
by geometrodynamics.
The objective of this review paper is to present a number of approaches in which gravity-like
features emerge (in different degrees of refinement and complexity) in various models, in specific
regimes. Rather than following a single line of thought, focussing on a single kind of models, or
a historical perspective, we will review a certain number of results trying to follow the logic flow
that will lead us from the most simple to the more complicated, questioning at each stage the
gains and the penalties to be payed. This is done in order to show the intricate web of relations
among the most different approaches, trying to establish relations among them and to make
easier the assessment of their pros and cons, but also disentangling the deep structural issues
from the accidental realizations that are due to the specific property of the model.
While the area of emergent gravity could be considered as a subject per se, it can be considered
a part of the research in quantum gravity, given that it might provide precious insights for the
problem of obtaining the correct semiclassical limit in a quantum gravity theory. In that context,
we will have to describe how the microscopic degrees of freedom are organized by their dynamics,
highlighting, when possible, the general properties of the state or of the regime leading, at large
distances, to a smooth Lorentzian manifold with quantum matter fields living over it.
For this purpose, the various techniques developed in the realm of (classical and quantum)
statistical mechanics, many body problem in the various incarnations encountered in condensed
matter physics, the renormalization group analysis and all the related ideas (recently, motivated
also from the systems relevant for quantum information) will be (most likely) decisive in the
elaboration of concrete strategies to solve the issue. This has already been shown in the concrete example of dynamical triangulations for Euclidean quantum gravity, of their Lorentzian
counterpart, causal dynamical triangulations [14, 15, 16, 17, 18], and of matrix models for two
dimensional gravity coupled to certain classes of conformal fields [85, 107].
We will not try to enter too much in the philosophical aspects of the idea of emergence, given
that the very definition is unclear. As a provisional definition, we will intend as emergence of
Emergent Models for Gravity: an Overview of Microscopic Models
3
a given theory as a reorganization of the degrees of freedom of a certain underlying model in
a way that leads to a regime in which the relevant degrees of freedom are qualitatively different
from the microscopic ones.
In absence of a unambiguous notion of emergence upon which a universal agreement can be
reached, we will avoid carefully to discuss its general principles, leaving them very vague, and
instead we will move directly to the discussion of the concrete models, proposed in the past
years to explain the long range phenomenon called gravity. We will focus on the merits of each
proposal, as well as the shortcomings. For different perspectives on the idea of emergent gravity,
the reader can see, for instance, [88, 89, 132, 133, 134, 135, 171, 195].
Disclaimer. The nature of this review is very specific. It is focused on the methods, ideas
and results of various models in which gravitational dynamics is recovered, more or less successfully, from a non-gravitational system. As a consequence, we will not provide a review of many
important results that refer to the thermodynamical behavior of gravity, that we have already
mentioned. These important results pertain to the thermodynamical layer of the description.
We will rather focus the attention layer immediately below, i.e. on the way the microscopic degrees of freedom get assembled to produce gravity, whose interest for quantum gravity is obvious.
For a discussion of the thermodynamical aspects of gravity, we refer to the extant literature.
This said, we will leave out of the analysis one relatively important topic on the emergence
of spacetime, the AdS/CFT scenario mentioned earlier. For this very important holographic
perspective we will refer, in addition to the references already given, to [23, 43, 78, 103, 130, 161],
where various different aspects are discussed, with special emphasis on the origin of gravity from
CFTs.
The plan of the paper is as follows. We will start in Section 2 by recalling simple ideas of
composite gauge bosons from fermion/antifermion pairs, and some key ideas of induced gravity.
In Section 3 we will briefly recall the main results of analogue models, which have provided
valuable inspiration for this whole approach. In Section 4 we will consider carefully the case for
Lorentz invariance, highlighting some important difficulties that are plaguing analogue models
with many components. In Section 5 we will give an account on how Bose–Einstein condensates
(BEC) can be used to have dynamical analogues of Newtonian gravity. While practically of
little interest, this allows a first concrete (actually literal) example of gravity as hydrodynamics
and of the emergence of the equivalence principle. We will pass then to the description of some
ideas related as the point of view that the graviton is a Goldstone boson associated to the
spontaneous breaking of Lorentz invariance, in Section 6, while in Section 7 we will examine the
possibilities that diffeomorphism, gravitons and their dynamics might emerge from a different
sort of phase transitions appearing in some lattice systems. We will also summarize the key
ideas that are behind the notion of “emergent gauge invariance”. In Section 8 we will then
describe some models in which there is no background geometry, where the metric genuinely
emerges as a composite field, while in Section 9 we will provide some examples in which the
signature of spacetime is determined dynamically. Concluding remarks and future perspectives
are presented in Section 10.
We will use the signature (−, +, +, +) unless otherwise specified, and we will use units ~ = 1,
c = 1, apart from the sections involving BECs, for which the usual nonrelativistic sets of units
are used.
2
A warm up
The question about the fundamental nature of the gravitational field, geometry and the spacetime manifold is just the most extreme case of the question about the fundamental nature of
the interactions (and of matter fields) that we observe. It is therefore interesting to start by
considering two very simple cases in which gauge symmetry and gravitational dynamics can
4
L. Sindoni
emerge. These two examples will represent two prototypes on which many proposals have been
based. Since they also allow us to discuss a number of important points, we start our tour from
them.
2.1
Bjorken’s model for electrodynamics
The idea that the fundamental interactions can be, after all, not so fundamental, is not a new idea
in theoretical physics. In a rather old work, Bjorken [47, 48] has shown how it might be possible
that a U (1) gauge interaction like electromagnetism can be generated in a purely fermionic
theory with a four fermions interaction. The idea is to start from Nambu–Jona Lasinio-like
fermionic systems [166, 167] described by the following generating functional1
Z
Z
i
µ
4
Z[Jµ ] = DψDψ̄ exp −
S[ψ, ψ̄] − ψ̄γ ψJµ d x
,
~
where the action is given by
Z
G
ν
2
4
µ
S[ψ, ψ̄] = d x ψ̄iγ ∂µ ψ − mψ̄ψ + (ψ̄γ ψ) ,
2
with G being the coupling constant associated to the four fermions vertex. Notice that this
generating functional allows us to compute correlation functions for the composite operator
Aµ = ψ̄γ µ ψ,
which transforms as a Lorentz vector.
Of course, this model is non-renormalizable by power counting, and hence it must be seen
as an effective theory valid at sufficiently small energies (with respect to the energy scale set
by G). A suitably defined Hubbard–Stratonovich transformation leads to the rewriting of this
in terms of the composite field alone
Z
2
Z
A
4
0
DA exp −i d x
Z[Jµ ] = N
− V (A − J)
,
2G
where V is the effective potential, and it is given, formally, by
Z
i d4 xV (A) = log det(iγ µ (∂µ + Aµ ) − m) − log det(iγ µ ∂µ − m).
By means of this integration over closed fermionic loops, the field Aµ inherits its own dynamics.
The lowest order effective action generated in this way gives
2
Z
2
1
λ
A
Seff = d4 x
+ 2 Fµν F µν + A2
,
2G 4e
4
where e is a coupling constant given by
1
1
Λ2
=
log
,
e2
12π 2
m2
with Λ an UV cutoff regulating the integrals (ultimately related to the energy at which the
effective theory breaks down).
1
Incidentally, NJL models were designed to reproduce some features of the BCS theory of superconductivity [38], in particular the formation of fermion-antifermion pairs, to be applied in the study of high energy
physics.
Emergent Models for Gravity: an Overview of Microscopic Models
5
The resulting effective theory for the composite vector field is not gauge invariant, since
there are terms containing A2 in the effective action. Nevertheless, the potential might lead to
non-vanishing vev of Aµ . In this case, the theory would give spontaneous Lorentz symmetry
breaking. The massless Nambu–Goldstone modes associated to this symmetry breaking might be
interpreted as photons, having the same kind of coupling to the fermions, i.e. minimal coupling.
For a detailed discussion, see also [46].
Such an example shows how it is possible to have composite vector mediators arising from
four-fermions interactions (see also [26, 90, 91, 213, 214]). In the past, models applying this kind
of reasoning to the case of the emergent linearized graviton have been put forward [22, 168],
implementing the idea that the linearized graviton might arise as a certain composite field
of a nongravitational field theory. However, the models have difficulties in going beyond the
linear order to include self coupling of the emergent gravitons, thus barring the way to regain
general relativity even in a perturbative sense (see, for instance [84] and references therein).
Furthermore, the question whether a four Fermi theory can be truly fundamental, and not only
an effective field theory valid below a certain scale, requires further clarifications2 .
Lorentz symmetry breaking is unavoidable, in this class of models. From a pure grouptheoretic point of view, the photon is described by a helicity one state. In absence of gauge
invariance, a mass term for a vector field cannot be forbidden. Therefore, one needs another
mechanism to keep the vector massless. Viewing the photon as a Nambu–Goldstone mode is
certainly an option. However, the price to pay is that it has to come from a vector theory,
and hence it has to break Lorentz invariance. Given the tight constraints on Lorentz invariance
violation (LIV), this option seems to require a lot of work on the fundamental model to be
really viable (essentially, a viable mechanism to suppress the LIV operators in the effective field
theory). We will come back to this point later.
There are other two key differences from a genuine gauge theory. First of all, besides the
massless Nambu–Goldstone modes, there are other polarizations for such composite vector fields.
They are massive, with mass determined by the shape of the potential V (A). If the potential is
steep enough in these components, they are not observable at low energy.
Second, the fact that a theory is gauge invariant implies Schwinger–Dyson and Ward–
Takahashi identities among the correlation functions, and, consequently, specific patterns in
the renormalization of the theory. These are the quantum version of the classical equations of
motion and of the conservation equations following from the presence of symmetries. A proposal
for the emergence of gauge invariance has to reproduce these identities, at least within certain
approximations that determine the regime of validity of the model itself.
Despite all these shortcomings, Bjorken’s model is a prototype for all the field-theoretic
models (i.e. described by some form of field theory on some notion of background spacetime
manifold) of emergent gauge systems as we know them. In this model, the gauge boson is
interpreted as a vacuum expectation value of a composite operator of certain fundamental fields,
whose dynamics generate a strong-coupling phase which is most effectively described by these
composite fields. In a sense, the resulting equations of motion (classical or quantum) might be
viewed as Landau–Ginzburg equation for the order parameter of the specific phase of the theory.
2.2
Induced gravity versus emergent gravity
Is it possible to generalize the mechanism to gravity? Before considering this very general point,
it is interesting to mention the mechanism of induced gravity. In the previous example, a central
2
There are mainly two problems. First, power counting argument suggest that the theory is perturbatively
nonrenormalizable. Second, these theories might saturate and then violate the unitarity bound for scattering
amplitudes. Of course, there might be a natural solution to these issues without invoking gauge invariance,
within the framework of asymptotically safe theories. However, the matter is not yet settled.
6
L. Sindoni
role is played by quantum corrections to the tree level action. It is instructive to see how these
can be relevant within a context in which gravity is introduced.
In [193], Sakharov suggested the idea that the dynamics of the gravitational field could be
seen as a manifestation of the “elasticity of spacetime” with respect to the matter fields living
over it. For a technical presentation and discussion of the various subtleties involved, see [222].
Assume to have a certain set of matter fields, denoted collectively by φ. Assume that they
propagate over a metric gµν which is, at least at a first glance, just an external field, without
any sort of dynamics. The only equations of motion relevant, at the tree level, are
2g − m2 φ = 0,
for the matter fields3 . However, quantum corrections significantly change the picture.
By simple manipulations of the functional integrals, it can be seen that loop corrections due
to the dynamics of the matter fields can generate an effective action for the gravitational field.
At one loop:
S1−loop [g] ≈ log det 2g − m2 .
The standard Schwinger–DeWitt technique [56] shows that this term involves a number of divergent contributions, which need appropriate counterterms to be renormalized. In particular,
among these terms there are what we call the cosmological constant term and the Einstein–
Hilbert action.
This picture has been further refined [2, 3, 245, 246], connecting the problem of emerging
gravitational dynamics with the breaking of scale invariance and the appearance of energy scales
separating UV from IR degrees of freedom. In the same spirit, but with a slightly different
twist, there are proposals of some models of gravity (and gauge interactions) entirely arising
from fermionic/bosonic loop corrections. These attempts are certainly very important in the
discussion of the nature of gravitational interaction, and must be taken into account. We will
devote an entire section to them later in this paper.
Induced gravity is certainly an option that should be considered. Nevertheless, it is perhaps
not appropriate to think of induced gravity as an issue of emergent gravity. Indeed, there is
a difference between induced gravity and models á la Bjorken: the mechanism described by
Bjorken is essentially the formation of a composite field, by some sort of fermion/antifermion
pairs condensation:
Aµ ∝ hψ̄γµ ψi.
Subsequently, the dynamics of the field Aµ is induced by the quantum corrections. The case of
induced gravity is fundamentally different in the first step: the pseudo-Riemannian structure
of spacetime is postulated a priori. It is assumed that the matter fields are living on some sort
of spacetime. What is not assumed is the fact that the metric tensor does obey some equation of
motion. Sakharov’s induced gravity, then, is a case in which only the gravitational dynamics is
induced by quantum corrections, and not the entire geometrical picture. Stated in another way,
in Sakharov’s picture, the graviton is not necessarily a composite particle (even if it is implicitly
assumed so).
In this sense, then, it is important to distinguish this scheme from a more radical emergent
picture, where all the concepts needed to speak about spacetime geometry are emerging from
a pre-geometric phase.
3
Of course, for fermions one should consider the corresponding Dirac equation and consistently take into
account the small differences with respect to the bosonic case. Small modifications are present for particles with
different spins. Nevertheless, the general result is unchanged.
Emergent Models for Gravity: an Overview of Microscopic Models
3
7
Emergent spacetimes: lessons from condensed matter
Despite the remarkable observational successes of GR, the gravitational field, due to its relative
weakness, remains very hard to be tested in regimes of strong curvature (where the behavior
of gravity significantly deviates from the Newtonian description and requires a fully relativistic
description) and simultaneously under experimentally controllable situations, where, at least in
principle, the separation between contamination effects and the physical effect is clear.
Besides the obvious importance of the strong curvature regime for the dynamics of the gravitational field itself, there are a number of kinematical effects that are relevant for the dynamics of
the matter fields living in the spacetime manifold. There is a first obvious question that arises,
in curved spacetimes, and it is the way in which matter fields feel the underlying geometry.
A feature such as nonminimal coupling should be more visible in strong curvature regimes (even
though it might have surprising impact in certain low curvature but large size systems [45]).
Besides the issue of the validity of the equivalence principle there are some enormously important phenomena, such cosmological particle creation and Hawking radiation, that have a huge
impact on our understanding of the origin of structures in inflationary scenarios, and on the
fate of unitary evolution in quantum mechanics when black holes can form [164].
Despite the fact that to date there are different ways to approach the theoretical aspects of
these effects, still, their status as proper phenomena is unclear. A direct detection is needed to
confirm the theoretical calculations. However, gravitational fields that are strong enough to lead
to particle creation phenomena are not available in a laboratory. With“true” gravitational fields,
the exploration of strong curvature seems to be limited to the mere observation of astrophysical
or cosmological phenomena, subject to all the related uncertainties and limitations.
However, it is possible to circumvent these difficulties, at least partially, and work with
“artificial” curved spacetime metrics even in a not particularly sophisticated laboratory. Indeed,
it has been realized by Unruh [216] that the pattern of propagation of waves in a moving fluid
strongly resembles the propagation of waves in curved spacetimes. The general idea behind
this fact is that, when a fluid is flowing with a given velocity field, sound waves are dragged
by the fluid flow. Hence, if a supersonic flow is realized in some region, sound waves cannot
go upstream. This is in analogy with what happens with trapped regions in general relativity.
Using this very simple idea it can be easily realized that some sort of sonic analogue of black
holes spacetimes can be realized.
This general expectation can be formalized in a rigorous mathematical theorem [221], which
shows how in some situations the properties of sound propagation can be exactly described by
a curved pseudo-Riemannian metric.
Theorem. Consider an inviscid, barotropic and irrotational fluid. Sound waves, i.e. linearized perturbations around a given fluid configuration, are solutions of a Klein–Gordon equation
for a massless scalar field living on a curved metric,


.
2 − v 2 ) .. −~
−(c
v
s

ρ 

.
·
·
·
·
·
·
·
gµν =

cs 
..
−~v
. I
3
This metric tensor is called the acoustic metric. It depends algebraically on the properties of
the fluid flow: the local density ρ, the velocity field ~v , and the speed of sound cs . Therefore, the
dynamics for this metric is not described by some sort of Einstein equations written in terms of
some curvature tensors. Rather, the acoustic metric is a solution of the fluid equations, i.e. the
continuity equation and Euler equation.
During the last years this result has been extended to other condensed matter systems, like
non-linear electrodynamic systems, gravity waves, etc.. For the present discussion it is important
8
L. Sindoni
to keep in mind that there are many condensed matter physics, from rather ordinary ones like
nonrelativistic fluids to more exotic ones like quantum fluids and graphene layers that can
exhibit a peculiar phenomenon: in certain conditions, to be specified according to the nature
of the system itself, the excitations, the perturbations or other signals do propagate, at least at
low energies/frequencies, over effective Lorentzian geometries (or with effective Dirac operators,
if fermions are involved), even if the original condensed matter system is relativistic under the
Galilean relativistic group of transformations. This result has been variously revisited, extended,
applied to many different systems with very different properties, and it has been developed into
a whole branch of research, investigating the properties and the possibilities of the so called
“analogue models for gravity”. Since this is not a review on analogue models, we refer to the
recent review [32] which presents an up-to-date picture of the investigation. For a detailed
discussion of 3 He models (and for a thorough discussion of the Fermi point scenario), see [229]
where some implications for emergent gravity scenarios are also discussed.
The bottom line is that one could imagine to realize effective curved spacetimes in systems
which could be studied experimentally (and not just observationally) in a laboratory, thus opening the possibility of detecting and testing directly some features of QFT in curved spacetimes,
like Hawking radiation and cosmological particle creation.
A crucial point must be kept in mind: these models are thought to be used to study effects
that involve just the fact that quantum fields live on curved spacetimes, the latter not being
necessarily solutions of some sort of Einstein equations. Indeed, these geometries are determined
by the equations of motion of the underlying condensed matter system, which have structural
differences with respect to normal diffeomorphism invariant systems of PDE that make a one
to one correspondence not possible, in general.
It should be said that analogue models are not just an exercise in mathematical physics. In
fact, analogue models can provide relevant insights and useful ideas about those situations where
spacetime is emergent from a complicated underlying quantum gravity dynamics. In the next
chapter it will be shown that the ideas we have already discussed in the area of analogue models
have a counterpart in many theoretical frameworks designed to describe the semiclassical limit
of quantum spacetime near the Planck scale, as well as their phenomenological consequences.
3.1
BEC systems
A particularly appealing system for the purposes of analogue gravity is represented by Bose–
Einstein condensates. A BEC is a particular phase of a system of identical bosons in which
a single energy level has a macroscopic occupation number. They are realized by using ultracold atoms (see [182]). Under these extreme conditions, quantum phenomena become a key
ingredient in determining the macroscopic properties of the system. The analysis of these
systems, therefore, opens a new area of investigation for quantum fields in curved spacetime
[33, 101, 102].
We do not want to enter a full fledged analysis of the theoretical understanding of the
physics of BECs. for which we refer to the extant literature. Rather, we recall a number of
basic facts that serve as a guide for the understanding of the physical content of the models
and their relevance. We do so, since many features will be common to other approaches. Also,
within the landscape of emergent gravity models, the BEC is perhaps the model in which the
transition between the microscopic domain and the long wavelength limit, displaying emergent
gravitational phenomena, is the cleanest.
The simplest description of a BEC system (specifically, for dilute Bose gases) is the mean
field description provided by the condensate wavefunction, a complex scalar (but can (and
has to) be generalized to higher spins) classical function encoding the fluid properties of the
condensate. The equations of motion for the many body problem can be translated into an
Emergent Models for Gravity: an Overview of Microscopic Models
9
effective equation for this function, the so-called Gross–Pitaevski equation, which is essentially
a nonlinear Schrödinger equation
i~
∂ψ
~2 2
=−
∇ ψ − µψ − κ|ψ|2 ψ,
∂t
2m
where m is the mass of the bosons, µ the chemical potential, κ controls the strength of the two
body interactions (the dominant scattering channel, in the dilute gas approximation), and ψ
√
is the condensate wavefunction. By means of the Madelung representation ψ = n exp(−iθ),
this complex equation can be split into two real equations, the continuity equation for the
number density n and the Bernoulli equation for the velocity potential θ. With respect to the
hydrodynamical equations for a perfect fluid, there is a correction term, known as the quantum
potential or the quantum pressure, that generates an additional contribution to the pressure of
the fluid. It is entirely quantum in nature, as signaled by the presence of ~ in its expression
Vq = −
~2 ∇2 n1/2
.
2m n1/2
A key role is played by the coherence length of the system, known as the healing length
ξ=
~
.
(2mµ)1/2
It plays many roles: it determines the typical dynamical scale of the condensate, as well as
the properties of the dispersion relation of the phonons. This latter, known as the Bogoliubov
dispersion relation, has the form:
ω 2 (k) = c2s k 2 +
~2 k 4
.
4m2
(3.1)
The healing scale controls the transition between the relativistic low energy dispersion relation
ω ≈ cs k to the high energy behavior ω ∝ k 2 .
In the low energy regime, where the quantum potential term is essentially inactive, we can
show that the phonons do propagate in effective curved acoustic Lorentzian geometries. Furthermore, the fact that in the case of BECs quantum properties are directly relevant for the
physics of phonons explains their relevance in simulating quantum phenomena in curved (acoustic) spacetimes. Indeed, the low energy effective field theory of the phonons is a quantum theory
of a scalar field on the acoustic spacetime defined by the condensate. Despite being a very simple
case, it is enough to reproduce, at least in principle, interesting phenomena of particle creation
in curved spacetime. For a recent and extensive discussion of BEC models, see [139].
The most important part in designing an analogue model is to control the physical properties
of the system itself in such a way to reproduce the desired acoustic metric. In the case of the
perfect fluid we have discussed above, the procedure is pretty straightforward: by controlling
the velocity profile one can obtain the various properties of the acoustic metric. In the case of
the condensate, an alternative way has been envisaged to achieve this goal. In the case of the
BEC the scattering length, related to the interatomic interactions, can be modified by acting
directly on the condensate, by means of the so-called Feshbach resonances [182]. By using this
technique, one can make the speed of sound a position dependent function. Therefore, instead
of keeping fixed the speed of sound and changing the velocity profile, in order to have subsonic
and supersonic regions it is enough to keep the flow’s velocity fixed and to change the speed of
sound.
The possibility of simulating black hole spacetimes and hence to have some sort of grasp on
the Hawking effect has been put forward since the works [101, 102] (see also [33]). Similarly,
10
L. Sindoni
it has been proposed that BEC could be used to realize analogue of expanding Friedman–
Robertson–Walker (FRW) spacetimes, and hence to have the possibility of check in a laboratory
phenomena which are typical of inflationary scenarios, like particle creation [34, 59]. In this
case, it becomes manifest how important is the possibility of controlling the scattering length in
realizing a curved acoustic metric.
The method of Feshbach resonance is so effective that one could realize physical situations of
rather exotic phenomena which do not have an analogue in the standard approach to quantum
field theories, namely, signature changes events [238]. By tuning the value of the scattering
length, and in particular making it negative, one can immediately realize that c2s < 0, which
means that the acoustic metric becomes a Riemannian one (signature (+ + ++)).
BEC-based analogue models, therefore, do represent a very nice opportunity to discuss many
different phenomena which are of interest for theoretical physics. There are two fundamental
gains from the study of these systems. First of all, testing phenomena of quantum field theories in curved spacetimes which have been only predicted and never measured directly, given
the difficulty of detecting them in strong gravitational fields generated by the astrophysical
sources, is an important achievement per se, making theoretical prediction subject to experimental verification (see also [139] for a dedicated overview).
An additional reason of interest is related to the so called trans-Planckian problem. In
the derivation of Hawking radiation and the spectrum of inflationary perturbations, one is
implicitly assuming that the effective field theory description (QFT in curved spacetime) is
holding up to arbitrarily high energies/arbitrarily small scales (for a discussion and references
see, for instance [159], but see [164] for a different perspective). This is certainly an untenable
assumption. Almost certainly, at sufficiently small scales, the EFT picture will break down,
higher dimensional operators will become more and more relevant up to the point that the
fundamental theory of spacetime (quantum gravity/strings/. . . ) might be required to perform
the calculations.
The trans-Planckian problem, then, could open the door to this (mostly unknown) physical
effects on low energy predictions. In this respect, it is important to understand how much the
predictions of Hawking’s spectrum and particle creation in an inflationary model are affected
by modifications to the UV behavior of the theory. In the case of BEC, we are in complete
control of the trans-Planckian physics of the system, i.e. the properties of the model below the
healing length.
As it has been shown, phonons do not have an exact relativistic dispersion relation: the
dispersion relation receives corrections which are more relevant the higher is the energy, with
the healing length scale playing the role of the separation scale between the relativistic branch
and the trans-phononic branch of the spectrum. The robustness of Hawking radiation and
particle creation against high energy modified dispersion relations has been discussed in several
works [28, 29, 30, 55, 77, 96, 217, 218]. Similarly, while a signature change event would lead
to an infinite particle production, the presence of high energy modifications of the spectrum
regularizes the divergencies, predicting a finite result [238].
The potentialities of these analogue systems are further highlighted by some works where
the issue of the backreation of the emitted radiation on spacetime (i.e. the precise way in which
Hawking radiation leads dynamically to the evaporation of the black hole) is explicitly considered [24, 25].
Recently, rather important progresses have been done on both the experimental side and
on the numerical side. In fact, very recently it has been discussed the realization of a BEC
analogue of a black hole [150]. Even though this is certainly encouraging, still this is not enough
for the purposes of detecting directly Hawking quanta. On the other hands, the production
of Hawking radiation from an acoustic black hole in a BEC has been detected in numerical
simulations [149], while experiments with other analogue models have shown signals of analogue
Emergent Models for Gravity: an Overview of Microscopic Models
11
Hawking radiation [41, 237]. Finally, some numerical estimates on the effects of cosmological
particle creation have been done in [235] (see also [185]).
It is clear that the underlying physics, for these analogue models, is non-relativistic: some
Lorentz-violating effects are gradually turned on as the energy is increased. Even though this
is a possibility, this might not be the case for “real” spacetime, and it might be that other kind
of UV completions of our low energy effective field theories would lead to unexpectedly large
effects on the low energy predictions of Hawking spectrum and cosmological particle production.
To decide this matter, of course, a complete calculation within a theory of quantum gravity is
needed.
3.2
Fermi point scenario
It is interesting to briefly mention here some important work done on fermionic systems. It is
hard to overestimate the importance of the discussion of analogue systems containing fermionic
degrees of freedom, given that these are among the basic ingredients of the Standard Model. As
a second instance, it gives intriguing hints that there might be a deep connection between the
particle spectrum, their interactions and topology, broadening then the perspective of emergent
systems.
The analysis of the excitations of fermionic system over the vacuum state involves the determination of the properties of the Fermi surface (even in a generalized sense) in momentum
space. Indeed, it has been shown that the properties of the excitations are essentially tied to the
topological properties of the Fermi surface [129, 228, 229] (see also [225, 227] for more recent
presentations). We refer to these references for more detailed accounts of the idea.
As a consequence, for fermionic systems the determination of the dispersion relations, and
hence of the emergent low energy spacetime symmetry, amounts to the determination of certain topological properties of the Fermi surface (in the generalized sense of the region of the
momentum space where the effective propagator has poles). This classification is robust, in the
sense that only topological reasonings are needed, and many details of the underlying condensed
matter system are simply not needed. This is of course a form of universality.
In particular, it has been established that the relevant universality class is the one associated
with the presence of Fermi points, in momentum space (we refer to the literature for all the important details). In this case, the inverse effective propagator, for an excitation of momentum pµ
reads
G−1 (p) = eµa Γa pµ − p0µ + corrections,
where p0µ is the position of the Fermi point in momentum space Γa are just the four Hermitian
2 × 2 matrices (I2 , σ i ) and eµa plays the role of a tetrad. In such a configuration, then, for
momenta close enough to the Fermi point, the energy spectrum is linear, and essentially one
recovers a notion of Lorentz invariance. The correction terms in the inverse propagator are
responsible for the restoration of Galilean invariance at high energy, but they can be neglected,
at low energy, as one can neglect the corrections to the relativistic behavior in the Bogoliubov
dispersion relation for the phonons in a BEC.
In fact, one can push a little bit more the analysis and show that it is conceivable to expect
that the matrix eµa and the vector p0µ become position dependent. In this case one has the
generalization of the result obtained for fluids and phonons in BEC: the excitation do propagate
in a nontrivial metric-affine background for momenta close to the Fermi point. Indeed, the
tetrad gives rise to a position-dependent metric tensor
g µν = eµa eνb η ab ,
12
L. Sindoni
with the connection being just a U (1) connection
Aµ (x) ∼ p0µ (x),
which might represent a form of electromagnetic field. The entire construction can be generalized
suitably in order to obtain the emergence of nonabelian gauge fields by choosing the correct
topology (i.e. universality class) of the quantum vacuum. This picture gives a strong suggestion
on the fact that the entire set of gauge interactions (gravity included) might be low energy
remnants of an underlying microscopic theory, with the low energy theory, its symmetries and
its dynamical content partially determined by topological properties of the quantum vacuum.
However, as in the case of BEC and, partially, of Sakharov’s idea, this is only half of the
problem. Indeed, this tetrad and these gauge fields appear as nondynamical classical background
fields: they are related to the collective fields that would describe the vacuum state, and they
would be determined by the same quantum equations of motions that determine the vacuum.
Hence, if we start from a nonrelativistic system containing nonrelativistic fermions, the equations
for these geometrical and gauge backgrounds will not be of the form of an Einstein–Yang–Mills
system.
Furthermore, as we will discuss briefly in the next section, in the case in which several species
are present there is no guarantee that at low energy there will be just one notion of effective
metric (see the discussion in Section 6.4.3 of [228] for this specific point). We refer again to the
papers and to the book mentioned above for additional references and extensive discussions on
the potential relevance of these ideas for an emergent theory of gravity, especially on the role of
the various energy scales involved.
4
The fate of Lorentz invariance
The examples discussed above suggests that Lorentz invariance4 that we observe and with which
we build the Standard Model, might be, after all, only a symmetry property associated to the
low energy/large distances regime that we probe with our experiments or observations. Instead
the fabric of spacetime at smaller and smaller distances could be characterized by a different
symmetry group, or by a totally different geometrical structure. One immediate manifestation
of this would be the phenomenon of dispersion for light rays, i.e. the difference in the speed
of propagation of light rays with different frequencies. The possibility that quantum gravity
effects might lead to nontrivial propagation of waves has now a relatively long story [11, 100],
and has flourished in what is called Quantum gravity phenomenology [19, 20, 131, 158].
Within the context of analogue models, the general procedure to derive the existence of low
energy Lorentz invariance involves the manipulations of the field equations (classical or quantum) for the system under consideration, the imposition of the appropriate boundary conditions
and, typically, a restriction of the analysis to a specific kinematical regime in which one or
more conditions are precisely met. This is because, while it is certainly possible to have Lorentz
invariance emerging in condensed matter systems, the result is not obvious at all.
4.1
Multi-component systems: normal modes analysis
So far we have presented results for which it is implicitly assumed that there is just a single field
describing the excitations. This may suffice to investigate phenomena like particle creation, as
we have discussed previously, given that the effect that is under consideration does not really
involve the kind of matter fields that are involved. However, it is important to understand how
4
We will often refer to the Lorentz group even though the Poincaré group should be mentioned instead. We
apologize for this abuse of terminology.
Emergent Models for Gravity: an Overview of Microscopic Models
13
generic is the emergence of a single pseudo-Riemannian metric when several components are
present. In addition to the obvious interest in understanding the conditions necessary to have
the emergence of a Lorentzian structure, it turns out that this analysis gives indications about
possible alternatives. It is instructive to discuss two cases, where the physical features are clear.
These will be paradigmatic for the cases in which more fields are involved.
The first case to be discussed is the propagation of light in a crystal. On scales larger than
the size of the cell, a crystal is an homogeneous medium: all its points are equivalent, having the
same physical properties. However, not all the directions are equivalent: in general a crystal is
anisotropic, the anisotropy being related to the preferred directions selected by the fundamental
cell.
Therefore, the propagation of signals, like sound waves, or light, is affected by the symmetry
properties. In the case of the propagation of light (for the propagation of sound waves see
for instance [151]), the Maxwell equations in vacuum are replaced by effective equations where
(see [152] for a complete treatment) the overall electromagnetic properties of the material are
included in the fields D, H, related to the electric and magnetic fields E, B by means of
constitutive relations, encoding the microphysics of the system in macroscopic observables.
To make a long story short, in the case of linear constitutive relations with negligible magnetic
properties (µij = δij ), the dispersion relation is provided by the so called Fresnel equation:
det(n2 δik − ni nj − εij ) = 0,
where εij encodes the linear constitutive relation relating the electric fields. This equation is
an equation for the refractive index vector n = k/ω. The shape of the space of solutions of
this equation depends on the shape of the tensor ε, which encodes the EM properties of the
medium we are considering. For instance, if εij = εδij we have an isotropic medium, and we can
see that n lives on the sphere of radius ε1/2 . The geometric interpretation is obviously given
in term of a single Riemannian metric, leading to a single pseudo-Riemannian structure for the
propagation of light signals, with the speed of light in vacuum replaced by the speed of light in
the crystal.
In the case of uniaxial crystals, we have that εij = diag(ε1 , ε1 , ε2 ). This leads to a Fresnel
equation which is factorizable:
n2x + n2y + n2z − ε1 ε1 n2x + ε1 n2y + ε2 n2z − ε1 ε2 = 0.
Therefore, this equation gives rise to two bilinear forms, with which one can define two pseudoRiemannian metrics, thus leading to a bi-metric theory. The two photon polarizations are
traveling at different speeds. This phenomenon is called bi-refringence.
In the case of biaxial crystals, we have three principal axis with three distinct eigenvalues,
the factorization of the Fresnel determinant into two quadratic terms is no longer possible: we
have to solve an algebraic equation of the fourth degree. In the three dimensional vector space
where the vector n lives, the surface defined by this equation is a complicated self-intersecting
quartic surface. The features of this surface are the origin of the interesting optical properties
of this class of crystals [54].
Notice the relation between the symmetry group of the crystal and the corresponding geometrical structure: the more anisotropic is the crystal, the more we break Lorentz invariance in the
analogue model. In the most general case, Lorentz invariance is not an approximate symmetry
in these analogue systems, even at low energies.
Another interesting class of analogue systems with a rather rich phenomenology is given by
the two components BEC systems [160, 236]. These are just a special case of the so-called spinor
condensates [215]. In these systems, there are two species of bosons, rather than only one as in
the BEC discussed previously. Similar considerations also apply to the Fermi point scenario, as
we have mentioned in the previous section.
14
L. Sindoni
These two components can be thought to be two different hyperfine levels of the same atom.
The two components are interacting, typically by laser coupling, which is inducing transitions
between the two levels. The analysis of the spectrum of the quasi-particles is considerably more
involved than the single component case. Nevertheless, in the hydrodynamic regime (when the
quantum potential is neglected), the spectrum can be obtained analytically. For the detailed
calculation, which is based on a suitable generalization of the Madelung representation, see [160].
The Hamiltonian describing the 2BEC contains a number of parameters: the masses of the
atoms, the chemical potentials, the strengths of the atomic interaction, and the coupling describing the induced transitions between the components. According to the values of these
parameters, the phononic branch of the spectrum shows different behaviors. In particular, one
can distinguish three geometrical phases: for generic values of the parameters, the analogue
geometry is not Riemannian, but rather Finslerian (see next subsection). If some tuning is
made, it is possible that, instead of this Finslerian structure, a bi-metric (pseudo-Riemannian)
will describe the propagation of the phonons (which are of two kinds, in these 2BEC systems).
Finally, if more tuning is performed by the experimenter on the various constants describing the
properties of the system, the two families of phonons perceive the same Lorentzian geometry at
low energy. In this specific corner of the parameter space Lorentz invariance as an approximate
symmetry is recovered at low energies.
Of course, this clean description holds only in the hydrodynamic limit, when the quantum
potentials are neglected. As in the case of single BEC, the higher the energy, the large will be the
contribution of the corrections to the dispersion relations (see (3.1)). Again, as we shall prove
later, dispersion can be seen as the manifestation of Finsler geometry. Therefore, in 2BECs
there are two ways in which we can see Finsler geometry emerge: first, by a generic choice of
the physical properties of the system, and secondly by the appearance of dispersion.
4.2
From analogue models to Finsler geometry
Analogue models have shown that it is certainly conceivable that Minkowski spacetime or,
more generically, Lorentzian geometries might arise in some effective regime of an otherwise
nonrelativistic system (in the sense of special and general relativity). This feature, however,
is not generic. If we take a general system with many components, extending the analysis of
the physical examples that we have considered, the normal modes analysis [35] shows that each
mode will have, in general, different speed of propagation. In the typical case, having n modes,
the dispersion relation can be expressed as
ω 2n = Qi1 ...i2 n k i1 · · · k i2 n .
In general this equation defines a (possibly self-intersecting) algebraic hypersurface of degree n in momentum space. Given the homogeneity of the relation under rescalings k → λk,
ω → λω, this hypersurface is nothing else than the generalization of the null cone of special
relativity.
The microscopic details of the system are encoded in the tensor Q. According to this, there
are different emergent geometrical structures. Tuning the microscopic parameters, it might
happen that the tensor Q factorizes into a product of metric tensors:
(1)
(n)
Qi1 ...i2 n = gi1 i2 · · · gin−1 in ,
In this case, the emergent spacetime structure is pretty clear: there is an effective multimetric
structure, where each mode is propagating on a different light cone determined by the corresponding metric structure. By additional tuning, it might be possible to reduce this multi-metric
structure to a single Lorentzian structure, when g (1) = g (2) = · · · = g (n) . We can say, then,
Emergent Models for Gravity: an Overview of Microscopic Models
15
that the emergence of an effective Lorentzian spacetime is not generic: some conditions have
to be satisfied in order for a single metric to be recovered. This is often translated in terms of
symmetries of the underlying model. For instance, anisotropies in crystals are directly related
to the appearance of birefringence.
Of course, even in the most symmetric case, the emergent Lorentz invariance is only an approximate symmetry of the spectrum. For instance, in condensed matter systems the underlying
spacetime symmetry is Galilean invariance, which has rather different properties than Lorentz
invariance. This fact is often contained into corrections to the lowest order/low energy equations for the perturbations. In systems like BEC, the quantum potential gets more important
the larger is the energy of the phonon, modifying the linear dispersion relation in the way described by the Bogoliubov dispersion relation (3.1). Dispersion is in general a feature that we
should expect in analogue spacetimes. Of course, dispersion is not compatible with a notion of
pseudo-Riemannian geometry.
However, the discussion made so far should have clarified that in analogue models the most
general situation is the one in which the tensor Q is not factorized. In this case, while the
dispersion relation is still homogeneous (hence there is no dispersion), it is generally anisotropic:
modes moving in different directions will propagate with different speeds. Again, this feature
cannot be described by Lorentzian geometry.
It is interesting that, despite that dispersion and anisotropic propagation do not have a direct interpretation in terms of Lorentzian geometry, they do have a geometrical interpretation
in terms of a suitable metric generalization of Riemannian geometry, namely Finsler geometry [27, 35, 192].
This is not the place to present a review on Finsler spaces. A rather exhaustive presentation of the most relevant technical points, with a list of references, is given in [197]. For the
moment it will suffice to say that a Finsler space is a manifold endowed with a (pseudo-)norm
in the tangent space of each point of the manifold. Clearly, Riemannian manifolds are special cases of Finsler spaces, being those Finsler manifolds where the norm is derived from
a scalar product. They can be also characterized in terms of symmetries: maximally symmetric Finsler spaces (i.e. those with the maximal number of independent Killing vectors) are
indeed maximally symmetric Riemannian spaces. In other words, Finsler spaces are in general
locally anisotropic metric spaces, while the locally isotropic ones are the ordinary Riemannian
spaces [1, 153].
Therefore, the appearance of Lorentzian or genuinely Finslerian structures in the effective
descriptions of emergent spacetime ultimately depends on the degree of isotropy of space, or
other, more general, symmetry requirements. Notice also that even in a spatially isotropic
situation with dispersion, the effective spacetime geometry is Finslerian [110].
Research on the concrete impact of Finsler geometry is an ongoing effort. The interested
reader can see [125, 187, 188, 189, 200, 201, 202, 203, 204, 219] for some concrete proposal
to define Finslerian structures in the Lorentzian signature to open the road for a concrete
investigation of the potential implications on phenomenology.
From a purely geometrical standpoint, Finsler spaces are perfectly reasonable geometrical
backgrounds to be used for the definition of distances and other metrical properties, as well as
for the propagation of particles and fields. However, from the physical point of view, anisotropic
Finsler spaces suffer from the problems of Lorentz violating theories in reproducing the observed
phenomenology.
The idea that spacetime structure at the Planck scale might deviate considerably from the
flat Minkowski background of special relativity is recurrent theme in models of quantum gravity.
Despite no definite proof exist that Lorentz symmetry breaks down in a quantum gravity model
(see however the already mentioned results [11, 100]), it is conceivable that near the Planck scale
spacetime starts deviating significantly from the continuum differential picture of Riemannian
16
L. Sindoni
geometry to become something different5 . Therefore, it is plausible that the Lorentz group
loses its preeminent role as well, to be deformed or replaced by other symmetry groups (see, for
instance, [111, 112, 113] and references therein).
One of the hypothesis that has been considered is that, due to the peculiar microscopic
dynamics of spacetime, dispersion relations (or, equivalently, wave equations) might receive
Lorentz-breaking corrections. The naive expectation is that, provided that the corresponding
operators are Planck-suppressed, Lorentz breaking effects might be invisible at low energy, in
which the theory flows towards a Lorentz invariant field theory. However, it has been shown
that this is not the case when quantum mechanics is included, in general. Indeed, Lorentz
violating operators at the Planck scale are able to contaminate low dimension operators (those
relevant at low energies) through radiative corrections [75, 137], leading to a naturalness problem
(essentially, the large hierarchy between the LIV scale and the scales involved in the physical
processes that we know so far, or other fine tuning problems) when this phenomenon is compared
to the experimental evidences, which put extremely strong constraints on Lorentz violation.
Despite some proposals, including supersymmetry [116, 165], the problem is still essentially open.
In absence of strong symmetry arguments (alternative to the direct imposition of Lorentz
invariance), multimetric geometrical phases or Finsler geometries are bound to emerge. A way
around this would be to get noncommutative spacetimes, as it has been suggested in the
past. However, this would amount to a much more radical departure from the usual setting
of differential geometry. The first lesson that we have to learn, when trying to emerge general
relativity from something else, is that even the kinematical setting, i.e. Lorentzian metrics, are
not automatically arising at low energies, and that a certain amount of work has to be dedicated
to the neutralization of all the possible sources of Lorentz symmetry violation6 . We will come
back to this point in Section 6.
Summarizing, in an emergent scenario, the mere emergence of the kinematical background for
gravity, i.e. Lorentz invariance, is not a trivial business at all, and might lead to an immediate
dismissal of the model from the very beginning.
5
Analogue nonrelativistic dynamics: the BEC strikes back
After having briefly exposed the main difficulties of getting the right kinematical framework
for gravity (i.e. having a properly defined low energy form of Lorentzian metrics) we can move
to the next level of difficulty: the emergence of some reasonable dynamics for the emergent
gravitational field.
Despite the fact that analogue models are designed to reproduce and study phenomena in
curved spacetime for which the exact form of the dynamics for the background geometry is
immaterial, this does not imply that no analogue dynamics at all can be reproduced.
As an example, it is possible to prove that in the presence of particular symmetries, the
equations of fluid dynamics can be mapped into the symmetry reduced equations for the metric
tensor. This has first been realized in [58], where the case of spherical black holes has been
considered. There, it has been proved that the gravitational field equations do have a precise
correspondence with the constrained steady fluid flow with an appropriate source.
Furthermore, it is easily realized that a mechanism like Sakharov’s induced gravity might
be at work in these models. Here, the loop corrections due to the quantum phononic field
might generate cosmological constant and Einstein–Hilbert terms. Nevertheless, there is a key
difference with the original idea of induced gravity: while in Sakharov’s proposal the dynamics
5
Of course, this is only one possibility, despite being the common one in traditional quantum gravity approaches.
6
See, on this issue, the Fermi point scenario discussed by Volovik [229]. See also [31] for a discussion of the
possibility that internal observers might not detect LIV, after all.
Emergent Models for Gravity: an Overview of Microscopic Models
17
of the gravitational field is produced entirely by means of loop corrections, in a BEC there is
already a dynamics for the condensate inducing a dynamics for the acoustic metric, i.e. the
Gross–Pitaevski equation. In these systems, then, the loop contributions are just subdominant
corrections [36, 223]. In this perspective, there have been proposals to use curved space techniques to assess the effects of backreaction of quantum fluctuations on the mean field description
of a condensate [24, 25].
BEC analogues do offer another intriguing possibility, that is to genuinely simulate nonrelativistic gravitational theories, like Newtonian gravity [109], and to discuss some structural issues
which are quite independent from the specific form of the gravitational theory at hand, like the
emergence of an equivalence principle [196].
5.1
Single BEC
In the following we will be very specific regarding the meaning of the analogue gravitational
dynamics. We will restrict the investigation to the equivalent of the nonrelativistic limit of
general relativity, i.e. the weak field, small speeds limit of Einstein’s theory. In this limit the
dominant effect onto the motion of particles is due to the Newtonian potential,
2ΦN = g00 + c2 ,
for massive particles. In the case of BEC (and other) analogues, this regime amounts to the
case in which the deviations from homogeneous static flow are small (ρ ≈ const, ~v ≈ 0).
The obvious difficulty to properly define a nonrelativistic model, the masslessness of the
phonons, can be circumvented by making them massive. This can be done very easily. Masslessness of the phonons in a BEC is a consequence of Goldstone’s theorem7 : the Hamiltonian of
a BEC is invariant under a global U (1) transformation of the second quantized field operators
creating and destroying particles, which is the symmetry related to the conservation of the total
number of bosons.
Breaking this symmetry, making the phonons pseudo-Nambu–Goldstone bosons allow for
a gap in the dispersion relation, i.e. a mass term. Of course this implies that a nonconservation
of the number of bosons has to be included. The origin of this might be an interaction with
a reservoir of bosons, if we think about atoms. However, one might also imagine that the
bosons which are condensing are quasiparticles or other collective excitations that might not
be associated to number conservation laws (magnons, spinons, etc.). In this cases, one has to
include in the equations of motion additional U (1) breaking terms.
In order to do so, the Hamiltonian Ĥ0 used for the number conserving system needs to
be slightly modified, by introducing a term which is (softly) breaking the U (1) symmetry, for
instance
Z
λ
Ĥ0 → Ĥ = Ĥ0 + Ĥλ ,
Ĥλ = −
d3 x Ψ̂(x)Ψ̂(x) + Ψ̂† (x)Ψ̂† (x) .
2
The parameter λ has the same dimension as µ. This is the most simple addition, relevant for
a dilute gas and for a system in which the number of bosons is conserved in average.
With this new Hamiltonian, the non-linear equation governing the field operators becomes
i~
~2 2
∂
Ψ̂ = [Ĥ, Ψ̂] = −
∇ Ψ̂ − µΨ̂ + κ|Ψ̂|2 Ψ̂ − λΨ̂† .
∂t
2m
These additional terms are responsible for the appearance of a mass term in the dispersion
relation of the phonon, which enables us to define properly a nonrelativistic limit. In particular,
7
For a detailed account of the concepts involved, the consequences and some applications, see [57].
18
L. Sindoni
the dispersion relation over a homogeneous condensate gives
4
1/2
p
2 2
2 4
E(p) =
+ cs p + M cs
4m2
where
M2 = 4
λ(µ + λ) 2
m
µ + 2λ
gives the “rest mass”8 , and
c2s =
µ + 2λ κnc
1+λ m
with
µ+λ
κ
the number density of the background condensate.
The second extension that has to be made involves the equations of motion. To properly take
into account the effect of quasiparticles moving over the condensate on the condensate itself,
the Gross–Pitaevski equation has to be replaced with the Bogoliubov–De Gennes equation for
the condensate wavefunction. This is obtained by using an improved mean field approximation
which includes the first effects of the noncondensed fraction. This is obtained by writing the
second quantized operator in the following approximate form
nc =
Ψ̂(x) ≈ ψ(x)I + χ̂(x).
The resulting equation for the condensate wavefunction reads
∂ψ
~2 2
=−
∇ ψ − µψ − λψ ∗ + κ|ψ|2 ψ + 2κn(x)ψ + κm(x)ψ ∗ ,
∂t
2m
n(x) = hχ̂† (x)χ̂(x)i,
m(x) = hχ̂2 (x)i,
i~
(5.1)
where we are neglecting hχ̂† χ̂χ̂i which is suppressed by the diluteness of the condensate and the
hypothesis of a small noncondensed fraction.
Notice that the form of this equation resembles structurally the semiclassical Einstein equations. However, the phonons are backreacting on the condensate not with their energy but with
their number density.
The final step is to consider the weak field limit, by restricting the condensate wavefunction
to the following regime
µ+λ 2
ψ=
(1 + u(x)),
(5.2)
κ
where u(x) 1 is a real function encoding the deviation from perfect homogeneity. One could
have introduced a small imaginary component (1 + u(x)) → (1 + u(x) + iv(x)). However, the
coupling of v(x) to the quasiparticles is weaker than the one of u(x). Furthermore, u(x) has
direct meaning in terms of the analogue Newtonian potential
1
3
ΦN (x) = − (g00 + 1) = − c2s u(x),
2
2
while the function v(x) is related, through gradients, to the shift vectors of the acoustic metric,
when written in the ADM form.
8
Notice the abuse of language: this term would just give the energy gap at zero momentum. We are using the
relativistic language in light of the analogy that has been established.
Emergent Models for Gravity: an Overview of Microscopic Models
5.2
19
Lessons: Poisson equation, mass and the cosmological constant
In the static limit, when we can neglect temporal gradients, the Bogoliubov–De Gennes equation (which is essentially the equation governing the hydrodynamics of the condensate) can be
rewritten in the following form:
1
2
∇ − 2 ΦN (x) = 4πGN ρmatter + Λ,
(5.3)
L
where we have defined the “matter” mass density as
1
ρmatter = M ñ + Re(m̃) ,
ñ = n − nΩ ,
2
m̃ = m − mΩ ,
with the nΩ , mΩ being the expectation values of the operators defined in (5.1) in the state
containing no phonons (i.e. with u(x) = 0), a vacuum term
1
2κ(µ + 4λ)(µ + 2λ)
nΩ + Re(mΩ ) ,
Λ=
~2 λ
2
an analogue Newton’s constant
GN =
κ(µ + 4λ)(µ + 2λ)2
,
4π~2 mλ3/2 (µ + λ)1/2
and the length
L2 =
~2
,
4m(µ + λ)
which corresponds to the healing length for this kind of condensate.
Therefore, with this simple models we can simulate a nonrelativistic short range version of the
gravitational interaction, obeying a modified Poisson equation. Unfortunately the range of the
interaction is controlled by the healing length, which is a UV scale, in this model. This signals
that the analogy is not good at all. However, this should not come as unexpected: indeed,
especially in effective quantum field theories, the mass of the bosonic fields (and hence the
range of the interaction they mediate) gets renormalized by contributions which are quadratic
and quartic in the cutoff of the theory, and hence pushed to the scale of the cutoff itself9 . At the
level of this semiclassical analysis adopted here this is not obviously at work, but the principle
is essentially the same: in absence of a custodial symmetry, there is no reason why should we
expect a long range/massless analogue gravitational field.
Of course, this is taken care of by diffeomorphism invariance, in general relativity. However,
we will see that in a multi-BEC case it is possible to have a long range interaction. Furthermore,
in more conventional scenarios for emergence of gravity-like theories, it is the nature of the
emergent graviton as a Goldstone boson associated to the spontaneous breaking of Lorentz
invariance to ensure the masslessness of the graviton.
Another interesting term is the vacuum term that we obtain in the effective Poisson equation (5.3). It is generated by purely quantum effects and it is related to the inequivalence
between the Fock vacuum of the original bosons and the Fock vacuum of the quasiparticles, as
its expression in terms of the Bogoliubov coefficients makes manifest [95, 109]. Interestingly,
a careful calculation shows that it is indeed a function of the so-called depletion factor, the ratio
9
Fermions would not suffer from this, provided that some additional custodial symmetry (e.g. chiral symmetry)
is active.
20
L. Sindoni
between the noncondensed phase and the condensed one. It is a measure of the quality of the
condensate: the smaller it is, the fewer are the atoms that sit outside of the condensate.
The relevance of this term is not only its nature as a peculiar quantum vacuum effect, but
also because it enters the effective equations of motion (5.3) exactly as the cosmological constant
enters the Newtonian limit in general relativity. Of course, this leads to the concrete possibility
to discuss one of the most excruciating problems of modern physics, which is the explanation of
the smallness of the observed cosmological constant.
Here, the smallness of the cosmological constant term would be explained by two basic observations [95]: first of all, its origin is not that of a vacuum energy, as it is normally taken for
granted in semiclassical gravity. Second, it is related to the depletion factor, which is in turn
related to the very possibility of speaking about a condensate. This might be an alternative
to the quantum field theory calculations of the vacuum energy, which are leading to the large
discrepancy between the theoretical prediction and the observed value of the vacuum energy
density [64, 175, 232].
Given that this model provides only an analogy, we cannot immediately export this lesson
to the case of the real cosmological constant, but it certainly constitutes another evidence that
the intuition of this term as a vacuum energy term might be fallacious, while more refined
considerations, often related to thermodynamic or statistical behaviors [9, 10, 140, 178, 206,
224, 226, 230], might lead to an explanation of its origin and, ultimately, its magnitude.
5.3
Multi-BEC, masslessness and the equivalence principle
There are several points that make the case of a single BEC a rather poor analogy for the phenomena that we normally associate to gravitation. First of all, there is only one kind of particles,
that make the model very poor on the matter side. Questioning the equivalence principle (in any
form), or the emergence of Lorentz invariance for all the particle species is impossible in principle. Second, there is the important issue of the range of the effective Newtonian gravitational
potential.
It is important, then, to extend the analysis as much as possible, as it has been done for
kinematical analogues, by going from single component systems to many components. This has
been done in [196], where the procedure that has been elucidated for the single BEC has been
extended to the case of spinor BECs, where several components are present.
We are not going to review all the steps of the model, which can be found in the mentioned
paper and do not add much to the discussion. Rather, we will focus on the outcome. The results
are similar to the ones for the kinematical analogues. In absence of symmetries,
• there is no Lorentz invariance at low energy, but rather multirefringence;
• there is no obvious analogue gravitational potential that is singled out, but a rather complicated pattern of interactions with all the components of the condensate.
The effective model that emerges, in the most general case, does not represent a viable
analogue for any sort of physically relevant regime of the gravitational interaction. One might
be pretty discouraged, at this point. However, interestingly enough, it is possible to cure all the
diseases with one simple recipe: imposing a global symmetry on the system.
In a system with N components, assuming an Hamiltonian which is invariant under permutations of the components among themselves, one can see that the following happens:
• there is no Lorentz invariance, but fields are organized in multiplets, within which a low
energy (multiplet-dependent) Lorentz symmetry holds;
• there is a single distinguished mode (associated to the simultaneous deformation of the
condensate wavefunctions) which plays the role of the Newtonian gravitational field;
Emergent Models for Gravity: an Overview of Microscopic Models
21
• the analogue Poisson equation, arising from the system of Bogoliubov–De Gennes equations, does not contain a term leading to a short range potential: the analogue interaction
is long range (or the graviton is massless);
• the various kind of massive phonons are coupled in the same way to the analogue gravitational potential: the equivalence principle emerges.
Notice that, despite the very simple class of models that we are playing with, we are still able
to discuss plenty of interesting features of realistic gravitational dynamics, like the range of the
interaction and the appearance of an equivalence principle.
This is teaching us an important lesson: in absence of strong symmetry arguments, the
emergence of a theory as simple as Newtonian gravity can be hopeless10 . In order to cure all
the possible diseases, symmetries have to be imposed on the microscopic model. This lesson is
of course extendible from this simple class of toy models to more refined scenarios. Nonetheless,
the BEC case can show basically all the potential difficulties that an emergent scenario has to
address, besides getting the correct Einstein equations.
6
Background Minkowski spacetime:
the graviton as a Goldstone boson
The analysis of analogue models has highlighted a number of basic points that might lead to
a failure, in the program of emergent gravity. The case of the BEC is instructive also at the
dynamical level. However, it was essentially a nonrelativistic model. One might wonder what
happens if we try to include special relativity in the game.
A simple point of view would be to consider gravitation as the interaction mediated by
a helicity two massless particle moving over a background Minkowski spacetime. Therefore,
before trying to emerge a form of background independent theory like general relativity, one
might want to attempt to describe a model in which a helicity two particle does propagate in
a flat background spacetime, allowing us to start to address the emergence of diffeo-invariance,
albeit only in a perturbative setting. This is the subject of the present section.
6.1
Comments on massless particles
General arguments of quantum field theory already tell us that if we want such a particle to
remain massless even in the presence of radiative corrections, it should be a Goldstone boson (if
it is not a gauge boson). In the case of the multi-BEC the gravitational potential was a kind of
Goldstone mode (even though it was associated to a discrete symmetry, not a continuous one) for
an internal invariance group that is spontaneously broken by the appearance of the condensate.
In absence of such symmetry breaking patterns, a custodial symmetry has to be introduced to
protect the mass term of the field encoding the emergent graviton from large renormalization
effects, allowing it to stay massless.
In addition to these caveats, we need to take care of some limitations that apply to Lorentzinvariant field theories containing massless particles. In particular, there is a well known
theorem11 , due to Weinberg and Witten [234], which is often presented as a crucial (fatal,
in fact [141, 142]) obstruction for a successful emergent gravity program. The theorem states
10
Obviously, one cannot hope to use the strategy of piling up all the matter fields into a single (super-)multiplet
of a “grand unified” symmetry group. The general reason is that, while gauge bosons belong to the adjoint
representation of the gauge group, the quarks and the leptons are in the fundamental one. This leads to severe
constraints [86] for the realization of a multiplet containing all the forms of matter fields that we know, and hence
for the opportunity to control Lorentz symmetry and the equivalence principle at once using this specific strategy.
11
See also [66].
22
L. Sindoni
precise limits for the existence of consistent theories with massless particles. It consists of two
parts, and it states that, in a Lorentz invariant theory, conserved currents associated to global
symmetries, as well as a covariantly conserved stress energy tensor, cannot contain, respectively,
contributions from particles with spin larger than 1/2 and 1. For a careful discussion of the proof
of the theorem, and for further references, see [162]. For additional comments, see [141, 142]
and Section 7.6 of [32].
The crucial ingredients for the proof of this theorem are a) Lorentz invariance and b) the
nonvanishing charges obtained from Lorentz covariant vectors and tensors. Indeed, the proof of
the theorem does not involve specific statements about the underlying Lagrangians, but only
rather general symmetry considerations.
As any theorem, it hinges on specific assumptions, which are rather easily overcome. For
instance, in case of the gauge bosons like the gluons and of the graviton the theory does not
apply, since the current for the gluons is not Lorentz-covariant conserved, and the graviton does
not possess a covariant stress-energy tensor (but rather a pseudo-tensor). In both cases, the
conservation law involves the gauge covariant derivative.
This theorem, nonetheless, poses rather strong constraints on the possible theories that can
be built in Minkowski spacetime. Of course one could say that gravity is not just the theory
of a spin-2 particle in Minkowski spacetime (not all the manifolds coming from the solutions
of general relativity are diffeomorphic to Minkowski spacetime). Nevertheless, it surely makes
sense to consider the linearized theory in sufficiently small neighborhoods. In this limit, then,
the theorem does apply.
With this caveat in mind, we can say that in an emergent gravity program this theorem
must be taken appropriately into account and appropriately evaded. There are (at least) two
“obvious” ways out:
• in a setup where a nondynamical background is present, allow for Lorentz symmetry
breaking, or
• make the spacetime manifold and its geometry emerge as well.
The first option is rather straightforward, and it is essentially what could be pursued within
scenarios like the one considered in analogue models, in which a preferred time function is
specified. However, this is also a (conceptually high) price to pay: a step back from Minkowski
spacetime to the notions of absolute space and time. Moreover, and most importantly, there is
the issue of recovering a low energy approximate Lorentz invariance: as we have already seen,
this is not at all an easy task without the requirement that the theory possesses some additional
symmetries or requires fine tunings which are able to realize this objective.
The second option is probably the most viable, conceptually appealing, but most demanding
in terms of new concepts to be introduced. If no reference is made to a background Minkowski
spacetime, but rather the graviton emerges in the same limit in which the manifold emerges, then
there is no obvious conflict with the WW theorem. Simply, what is called the gauge symmetry
in terms of fields living on the given spacetime is the manifestation of an underlying symmetry
acting on the fundamental degrees of freedom, in the limit when they are reorganized in terms
of a spacetime manifold and fields (gauge fields and gravitons in particular). We will be more
specific on this special class of models in the second part of this paper.
The bottom line of this very concise overview of the WW theorem is clear: to obtain a realistic
model of emergent gravity one must ask for very special mechanisms to be at work in the model,
to bypass what would be rather compelling obstructions to the recovery of gravity even in the
weak field limit.
The WW theorem is part of a family of theorems that have been established on the limitations
that have to be respected to produce theories contained interacting fields with high spins.
However, this topic is still under consideration: in the research area of higher spin field theories
Emergent Models for Gravity: an Overview of Microscopic Models
23
there is an effort to develop models of interacting field theories which are Lorentz invariant,
gauge invariant, and containing spins higher than 2, thus avoiding the various no-go theorems
on massless particles. For a pedagogical review, see [39].
6.2
Gauge invariance from spontaneous Lorentz violation
After this long preliminary discussion, we will briefly describe the idea of evading the WW
theorem by means of Lorentz violation. We will assume that there is a background Minkowski
spacetime, and that there are some fields defined over it. As it has been discussed in the case
of Bjorken’s model for emergent electrodynamics, it is possible that QED and the other gauge
interactions can actually be simulated by non-gauge theories in which the mediators of the
interactions are massless Nambu–Goldstone bosons. The idea is rather old [46, 74, 119, 120],
but in recent years the subject has been reconsidered [70, 71].
In order for the Goldstone modes to have the correct coupling to conserved currents or to
the stress energy tensor, they must have vectorial (or tensorial) nature. This implies that they
should arise as the excitations around a ground state of a theory for a vector or for a tensor field
possessing a potential term which allows for spontaneous symmetry breaking. The prototype of
such a behavior is the bumblebee model [50, 51], whose Lagrangian is
1
µ2
λ
L = − Fµν F µν − Aµ Aµ + (Aµ Aµ )4 ,
4
2
4
where µ has the dimensions of a mass and λ is a dimensionless coupling. The classical equations
of motion read:
∂µ F µν + µ2 Aν + λAµ Aµ Aν = 0.
The ground state for such a theory has infinite degeneracy, which can be labeled by constant
vector fields Aµ satisfying
Aµ Aµ = −
µ2
.
λ
If we introduce a (timelike) constant vector field nµ with unit norm and write Aµ = M nµ , we
have that M = µλ1/2 . These vacua break Lorentz symmetry, and hence we expect Nambu–
Goldstone bosons to arise. Of course, the resulting theory seems to lead, almost unavoidably,
to observable Lorentz violation. In fact, there is more. In [69] it has been shown that, if
we tune the parameters of the model in such a way that, even though spontaneous symmetry
breaking of Lorentz violation occurs, its effects (essentially in the scattering amplitudes predicted
by the theory) are not visible, the model thus defined displays also local gauge invariance as
a byproduct.
Concretely, this means that, when considering fluctuations around these backgrounds
Aµ = M n µ + a µ
the polarizations of aµ corresponding to the massless NG modes are coupled to the conserved
current in such a way to mimick Lorentz invariant electrodynamics in a noncovariant gauge.
Technically, this can achieved whenever the S-matrix for the model has the properties of
being transverse, i.e. for a process containing an external aµ line with momentum q µ and polarization µ ,
S(q, p) = µ (q)Mµ (q, p),
24
L. Sindoni
where p denotes collectively all the labels of the other states involved in the process, obeys
Mµ (q, p)q µ = 0.
In this case, the longitudinal polarization of aµ , µlongit ∝ q µ , effectively decouples from the
scattering involving conserved currents. Of course, this implies that a transformation
µ (q) → µ (q) + iα̃(q)q µ ,
or, in real space
aµ (x) → aµ + ∂ µ α(x),
becomes a symmetry of the scattering amplitudes.
Within this framework, spontaneously broken Lorentz symmetry, together with the (un-)observability criterion, leads to the emergence of local gauge invariance as an accidental symmetry
(in the technical sense of the term, see [233, Section 12.5]).
This picture has been variously extended to the case of gravity [44, 50, 51, 72, 73, 146, 148,
168, 183]. For this more exotic case involving tensor fields, the background configuration is
identified via conditions of the form
Hµν H µν = ±M 2 ,
where M is again an unspecified model-dependent energy scale associated to the UV physics responsible for the effective model considered. Clearly, the solutions to these equations do identify
background configurations Aµ or Hµν that, despite translational invariance, break a number of
generators of the Lorentz group. According to Goldstone’s theorem, we should expect an appropriate number of massless modes, associated to the number of broken generators. Now, these
Goldstone bosons can be coupled to the other fields in such a way to mimic the pattern of gauge
invariant field theories, in a way similar to the case discussed for the photon and the nonabelian
gauge fields.
Similarly to the case of the photon, in this case, to ensure the effective gauge invariance of
the emergent graviton, i.e. the decoupling of the other polarizations, it is enough to ensure that
the scattering amplitudes involving external gravitons are transverse,
S = µν (q)Mµν (q, p),
kµ Mµν = 0.
Under this hypothesis, the scattering amplitudes of matter fields (albeit not the Lagrangian)
are invariant under the gauge transformations
µν (q) → µν (q) − i kµ ξ˜ν + kν ξ˜µ ,
or, in real space,
hµν (x) → hµν (x) + (∂µ ξν + ∂ν ξµ ) .
This ensures that the additional polarizations of the field hµν not associated to the helicity two
graviton effectively decouple from the conserved stress energy tensor of the other particles, not
participating to what we would call gravitational scattering. In this sense, then, one would have
the emergence of diffeomorphism invariance at the linearized level. For a discussion about the
interplay of Lorentz invariance, gauge symmetry and scattering amplitudes, see the discussion
in [231].
Emergent Models for Gravity: an Overview of Microscopic Models
6.3
25
Limitations
While the construction of these models show that it is certainly possible to generate at least
at the classical level something that resembles gauge interactions (gravity included), there are
a number of questions that need to be addressed.
First of all, clearly, these results are perturbative in nature, being based on the expansion of
the models around fixed backgrounds, viewing the graviton as a linear perturbation around the
given background
gµν = ḡµν + hµν ,
with gauge symmetry represented by the action of infinitesimal diffeomorphisms (i.e. involving
the part of diffeomorphism group that is connected with the identity, and, in fact close to it in
an appropriate sense),
hµν → hµν + ∇µ ξν + ∇ν ξµ
(6.1)
with the only effect of Lorentz violation being equivalent a specification of a noncovariant gauge
condition.
These model have to face the challenge of going beyond the linearized theory, including, for
instance, the self-interactions of the gravitons with themselves in a way that is consistent, at
low energies, with the expansion of the Einstein–Hilbert action (or more general actions). This
problem stands out even at the classical level. At the quantum level, the challenge will be to
reproduce the Ward identities, conservation of currents, and, in general, all the consequences of
diffeomorphism invariance on the quantum theory. These will be the true test for these models.
A more concrete, worrisome problem is related to the percolation of Lorentz violating effects
from high dimensional suppressed operators to low dimensional ones, a phenomenon that happens to be generic in quantum field theories with Lorentz violation. The results on the generation
of gauge invariance from Lorentz breaking have to be confronted with the presence of radiative
corrections to tree level results. While it is certainly possible, by tunings, to resolve the issue, on
general grounds, radiative corrections will introduce visible effects even at low energies [65, 148].
For instance, in the case of the graviton it can be argued that radiative corrections will make
the dispersion relation depend on the background vev for the tensor field
k µ kν (ηµν + aHµν ) = 0,
where a is a computable coefficient. The effect is visible at low energy, involving the part of the
kinetic term with only two derivatives, and leads to direction dependent speed of propagation
according to the symmetry properties of Hµν . For extended discussion on the phenomenology
of this class of models we refer to the literature [50, 65, 147, 148].
However, the most serious drawback of these models is conceptual, and not practical: what
is the origin and the dynamical principle giving origin to the background Minkowski spacetime?
This latter appears to be a rigid structure which is not governed by equations of motion. Even
though it is possible to have a special relativistic framework, clearly a general relativistic picture
is incompatible with the presence of such a construction, which pinpoints preferred classes of
reference frames/coordinate systems which are adapted to the presence of the background.
7
Towards pregeometry
The models presented so far, while highlighting a number of interesting possibilities, are quite
unsatisfactory when confronted with the pillars of general relativity (or its extensions). One
of the most distinguished features of the latter is the invariance of the physical content of the
26
L. Sindoni
theory under the action of diffeomorphisms. There are two senses in which this is relevant. A
more conceptual one, which is related to the fact that a theory should not make reference to rigid
background scaffoldings, when dealing with its predictions, whose nature should be essentially
relational. The second reason is more practical, as it strongly influences also the dynamics of
the theory itself. Indeed, loosely speaking, one can view general relativity as a sort of gauge
theory for the diffeomorphism group, and the invariance under this group dictates the way in
which the various matter fields should be coupled to geometry and among themselves.
In fact, diffeomorphism invariance can be shown to lead also to a specific restriction of
the form of the dynamical equations for the gravitational field, which has to be governed by
constraints obeying essentially a Dirac algebra [127].
In more mundane terms, this means that the gravitational field, when expanded around flat
spacetime, can be described in terms of a Pauli–Fierz Lagrangian for a helicity two particle,
coupled to the SET of the other fields. Furthermore, diffeomorphism invariance selects also
the possible continuations to the nonlinear regime (given that the action should be specified in
a diffeomorphism invariant way).
Interestingly enough, diffeomorphism invariance is also another way to evade the WW theorem, given that, for the graviton, it is not possible to define a (Lorentz invariant) covariantly
conserved stress energy tensor, the energy of the gravitational field being contained in a pseudo
tensor.
7.1
Spin systems and quantum graphity
It is worth to mention some approaches where the graviton emerges as a collective mode exploiting certain forms of phase transitions which do not make use of order parameters, thus promising
to give rise to diffeomorphism invariance not through conventional symmetry breaking.
As we have said earlier, one could interpret the emergence of diffeomorphism invariance as
the approximate validity of the Hamiltonian constraints. In this sense one might be tempted to
construct an emergent model for quantum gravity for which the constraint surface of the classical
phase space is just the location of the minima of some very steep potential. If this happens, there
is room in the model to obtain a helicity two mode that can be formally described by a rank two
symmetric tensor field with gauge symmetry given by (6.1). We have already reviewed some of
the approaches supporting this idea. Here we want to briefly mention some recent attempts to
achieve this by means of suitably designed lattice/spin systems.
The models of emergent gauge symmetries that we were discussing previously are all based on
the exploitation of the field theoretical properties of phase transitions characterizable in terms of
Landau–Ginzburg theories for certain tensorial order parameters. However, these do not exhaust
the list of possible phase transitions. Recently, it has been realized that there are interesting
(quantum) phase transitions that cannot be characterized by order parameters associated to
local field theories. Rather, they are characterized by topological properties of the ground state
of the model. They are part of the subject known as topological order [239].
Within this context, a prominent role is played by local bosonic models: bosons sitting on the
nodes of lattices, whose dynamics is described by certain properly designed Hamiltonians. It can
be shown that, under certain conditions, the models undergo specific forms of phase transitions,
called string-net condensation, whose results are rather striking. Indeed, the analysis of the
excitations above the ground states shows that they are indeed described in terms of fermions
interacting via an emergent U (1) gauge interaction [154, 155, 156, 240].
This is certainly an important result that is undergoing rather intense investigations. Interestingly enough, concepts of topological order, as well as of string net condensations have been
shown to bear strong resemblance with spinfoam models for quantum gravity [87, 181]. Besides
this, local bosonic models have been considered to understand whether it is possible to emerge
Emergent Models for Gravity: an Overview of Microscopic Models
27
the graviton, within this class of models, as a collective excitation around a string-net condensed
state. To this purpose, it has been shown that it might be possible to design Hamiltonians leading to the desired result [117, 118]. We refer to these papers for the technical details of the
construction of the models and the methods to extract the physical content.
Despite the obvious interest of this outcome, there are some basic difficulties that, to date,
have not been overcome. First of all, these models are based on fixed nondynamical lattices,
thus leading to worrisome features that require some additional care. The first practical inconvenience is that a fixed nondynamical lattice for sure breaks Lorentz symmetry. Therefore, as
we have already discussed, the recovery of a low energy notion of Lorentz invariance will require
the imposition of a custodial symmetry, or, in the worst case, a fine tuning in the Hamiltonian.
Similarly, the presence of a Hamiltonian might suggest that there is also a preferred notion
of time, i.e. the lapse function has a fixed value and does not represent a Lagrange multiplier
associated to the Hamiltonian constraint of reparametrization invariant theories like general relativity. Finally, the background nondynamical lattice constitutes a violation of diffeomorphism
invariance/background independence in the hardest possible way. Therefore, extra polarizations
of the graviton have to be expected and have to be made massive by hand, with a mass scale
whose origin is not clear.
A way to attack these problems is suggested by known examples of models of quantum gravity
with fundamental discreteness, like (causal) dynamical triangulations [14, 15, 16, 17, 18], causal
sets [53, 126], and considerations within loop quantum gravity [191], spinfoam models [181] and
group field theories [99, 170, 172, 173]. All these approaches indicate that an obvious way out
should be a dynamical random lattice. While the “dynamical” part is still essentially work in
progress, on which we will comment later, the random nature of the lattice might be able to
save Lorentz (or Poincaré) invariance, at least in a statistical sense, without leading to preferred
frame effects or modified dispersion relations [52].
An attempt in this direction which is treatable with available techniques is represented by the
so-called quantum graphity models [143, 144, 145] (see also [60, 61, 76, 122] for related work).
The idea is very simple. Instead of starting with a given finite lattice with N nodes, which
consists of sites and links connecting the sites, one can imagine to start with its progenitor, the
complete graph with N nodes. The lattice structure emerges dynamically when an appropriate
Hilbert space is associated to this graph, in which each state corresponds to a particular connectivity configurations in which links are turned on or off. Some of these configurations might be
interpreted in terms of regular lattices in certain dimensions (which is variable, in these models),
while others may not have this interpretation.
The Hamiltonian, and the partition function describing the statistical mechanics of the system, are constructed in such a way to obtain a macroscopic notion of space (and of its geometry)
only in a statistical sense. In particular, the investigation of the statistical mechanical properties
of the models tends to show the existence of at least two phases: a high temperature phase in
which the state of the system is dominated by graph configurations which are highly connected
and do not have an interpretation of a nice smooth geometry, and a low temperature phase in
which it is possible to recognize the pattern of smooth regular geometry. The phase transition
separating the two phases has been called geometrogenesys.
While numerical methods are the main road to the content of these models, a mean field
approximation technique has been developed [61], which makes use of a mapping of the models
to particular spin systems similar to the Ising model, offering the possibility of an analytic study,
within the given approximation scheme.
Of course, establishing the existence of a phase in which a macroscopic notion of geometry
can be given is only part of the problem. The emergence of the right form of gravitational
interaction is still work in progress. However, interestingly enough, it is possible to see that,
in certain cases [60], these models do possess states associated to peculiar graphs in which the
28
L. Sindoni
excitations do propagate as they would do in a continuum spacetime geometry with a trapping
region (ideally, a black hole).
These kind of models are very intriguing, but they are not free of ambiguities. First of all, the
construction of the Hilbert space first, and then of the Hamiltonian suffers from ambiguities.
Lacking a notion of geometry with which operators can be classified into UV or IR, a systematic
construction of the Hamiltonian following the rules of effective field theories appears to be
impossible. Given the fact that a complete graph is used at the foundation implies that the
emerging geometry could possess any dimension (actually, any number from 0 to N ): this is
certainly an ambiguity that requires some care. Finally, given that the model is presented with
a proper Hamiltonian, the status of Lorentz and time reparametrization is unclear: Lorentz
invariance violation might be present, the long range continuum limit being similar to Lifshitztype theories [128], or just be a spurious effect due to the fact that the definition of the model
corresponds to the choice of a not covariant gauge. Apart from the dimensionality problem,
many of these issues are affecting the class of random models that we presently know: for
instance, the imposition of causality onto random matrix models [42] seems to point in the
direction of having preferred foliation effects, or, at least, nondynamical structures.
7.2
Emerging dif feomorphisms
At this point it is clear that the next kind of questions that we need to address, in our path for
the construction of a theory of emergent gravity, is whether diffeomorphism invariance can be
made to emerge as well. There are two scenarios in which this question can be addressed.
The first one is obvious, and will be the subject of the next sections: there is no spacetime
manifold, or, at most, a nondynamical topological manifold over which no metric or connection
structures are specified. In this case diffeomorphism invariance should be guaranteed by the
absence of background absolute structures [114], which forces the observables and the dynamical
structures to be purely relational. We will discuss some of these models in the next section.
The second scenario is the one in which there is a background spacetime with a given geometrical structure, nondynamical, providing a scaffolding, and diffeomorphism invariance is just an
emergent concept, valid only in a certain corner of the phase space of the theory at hand.
Essentially, the key idea that is behind emergent diffeomorphism invariance has already
become apparent in the case of the graviton as a Goldstone boson. Diffeomorphism invariance
can be considered to have emerged whenever the form of the theory at hand can be made to
coincide with a gauge fixed version of diffeomorphism invariant theories in the same regime.
Concretely, one can picture the emergence of gauge invariance in a specific corner of the
phase space of a theory as the fact that, in correspondence of that specific regime, the dynamical
equations of motion contain a potential term which becomes so steep to be able to approximate,
for all the practical purpose, a Dirac delta enforcing the constraints characterizing the canonical
description of a true gauge theory.
Notice that there is a difference with the Stückelberg method of introducing auxiliary fields to
make a theory look gauge invariant. Here we are making a statement about the physical content
of the theory, having in mind the direct comparison of the physical quantities as computed in the
two frameworks. Emergence of a diffeomorphism invariant theory would be then qualified as the
property of all the computable physical quantities to be in agreement (within approximations)
with those computed, in the appropriate regime corresponding to the specified background, in
a diffeomorphism invariant theory.
Concretely, if all the processes of a given model, like the scatterings mediated by what
would be identified as the mediator of the gravitational interaction, do reproduce (exactly or
approximately) the same results obtained within diffeomorphism invariant models, expanded
around the same background defining the regime of the theory that has to be compared to, the
Emergent Models for Gravity: an Overview of Microscopic Models
29
empirical content of the two theories can and has to be considered equivalent (see, on this, [31]
and references therein for the discussion of the case of Lorentz invariance).
Apart from the case of the graviton as a Goldstone boson scenarios, discussed in the previous
section, there is only the case, to the best of our knowledge, studied in [108], of the emergence
of diffeo-invariant Nordström theory for conformally flat gravity (and of Lorentzian signature)
out of a theory defined in a flat Euclidean background. Obviously, such studies do make sense
only in a perturbative, semiclassical regime, for which a notion of spacetime manifold, with its
geometry, can be specified. A comparison with the full quantum gravity regime of background
independent theories would not be appropriate.
Of course, the emergence of diffeomorphism invariance, in these regimes, would suffer of the
same problems of the emergent low energy Lorentz invariance: being an approximate symmetry,
it will be possible to discover specific signatures of the violation of the conservation equation
associated to it.
There are at least two kind of effects that one has to expect. First of all, given that
diffeomorphism invariance implies the covariant conservation of the SET, ∇µ T µν = 0, the most
immediate place to look for would be the total matter energy balance in gravitational processes,
looking for a nonconservation of the SET
∇µ T µν 6= 0,
albeit this line of thought might be leading towards a confusion with the problem of dark matter
(or more general dark components of the matter content).
Another obvious direction to be pursued would be the detection of extra modes for the
gravitational field. In linearized general relativity, local Lorentz invariance and diffeomorphism
invariance reduce the 10 components of the metric tensor to two propagating degrees of freedom,
describing gravitons. The other components are essentially unphysical gauge modes. However,
a breaking of either Lorentz invariance (which accompanies these models as well, generally) or
diffeomorphism invariance would promote some of the would-be gauge modes to the status of
physical modes. These modes would be extremely dangerous for the compatibility of the theory
with the observations, given that, on general grounds, their coupling to the matter SET would
be as strong as the one of the emergent gravitons. However, the steepness of the potential might
be of help in making these extra polarizations extremely massive, so that it will be extremely
difficult to excite them. While this is certainly a solution, it opens a naturalness problem related
to the magnitude of the energy scale M controlling the extra polarizations to the other scales
of gravity and particle physics.
Of course, quantum mechanics has to be appropriately taken into account. In addition to the
previously mentioned problems, radiative corrections might lead to dangerous corrections which
would make the models not viable. For instance, the mass term for the would be graviton could
be subject to large renormalization effects, given that the symmetry that would able to protect
it, diffeomorphism invariance, is not present. More work is needed, in this direction. A first
attempt to define an effective field theory which would be able to systematically address the
phenomenology of the breaking of diffeomorphism invariance is presented in [21], to which we
refer for further readings.
For all these reasons, models of emergent gravity with a background metric structure have to
be considered with particular care, and might not be considered as completely satisfactory and
physically viable models of emergent gravity.
8
Background independent pregeometrical models
The discussion has brought us to the last step: emerging geometry from pregeometrical degrees
of freedom. It is often stated that, on general grounds given by considerations of classical and
30
L. Sindoni
quantum field theories defined over Minkowski spacetime (e.g. the Weinberg–Witten theorem),
that a model of emergent general relativity will generically bear signs of Lorentz symmetry
violation and diffeomorphism symmetry breaking.
Of course, this is a simplistic point of view due to the particular starting point (that might be
circumvented, as we have discussed), that is field theories on some form of metric background.
There are very general difficulties in defining working models for emergent gravity which do
not make reference to such background structures, but this does not means that there are no
models at all.
To define diffeomorphism invariant quantum field theories it is necessary to define appropriately a volume element with which to define, in an invariant way, an integration over the
(background) manifold12 . The role of the volume element, in the case of general relativity, is
played by the square root of the modulus of determinant of the metric (or by the determinant
of the vielbein in a first order treatment). In absence of these structures, something else has to
be proposed13 .
The models we are going to describe (which are related one another) are formulated in such
a way to provide, first of all, a proper definition of volume element. Then, the metric is obtained
as a collective field, with its dynamics induced along the lines proposed by Sakharov. However,
they are not merely a form of induced gravity, where the metric tensor only acquires a dynamical
term. Rather, they promise to deliver the metric tensor itself.
The class of models we are going to discuss have been first formulated in a series of papers
[5, 6, 7, 8, 212], in which Sakharov’s idea is implemented in a model where no explicit notion of
geometry is introduced at the beginning. Rather, the metric tensor (or the vielbein/vierbein in
a D/4 dimensional first order formalism) is interpreted as arising from a nonvanishing vacuum
expectation value of certain composite operators defined in diffeomorphism invariant quantum
field theories. In the following we are going to discuss the models at length, given that they
represent interesting and significant alternatives to the scenarios that we have described so far,
being able to be pregeometric without losing contact with the language and the concepts of
quantum field theory.
Consider a multiplet of scalar fields φi , i = 1, . . . , N , whose classical action is given as
Z
1/2
S[φ] = d4 x − det ∂µ φi ∂ν φj δij
F (φ),
(8.1)
where F is an arbitrary potential. Despite this action might appear to be rather exotic, in
two spacetime dimensions is just the Nambu–Goto action for the string embedded into an N dimensional spacetime, with embedding functions given by the fields φi . In higher dimensions,
it represents the action of a membrane embedded in a pseudo-Riemannian space.
A partition function for this kind of action is easily written
Z
Z = D[φ] exp(−iS[φ]).
Clearly, such a partition function is difficult to be defined rigorously, at least in the bosonic
case. The general idea to get around this difficulty is to use a semiclassical treatment. Indeed,
12
Here the manifold might be considered as given or not. The important point is that it is just a topological
manifold, without any preassigned metric structure.
13
Interestingly, in Schrödinger book [194] on general relativity it is mentioned the possibility of a purely first
order action for pure gravity that does not make reference to a metric, but only to a connection. The action reads
Z
d4 x (det(Rµν (Γ)))1/2 .
Of course, the role of the metric is not just to provide us a notion of connection and curvature, but also to define
the kinetic terms for the matter fields. It is not clear how to couple matter to this kind of models.
Emergent Models for Gravity: an Overview of Microscopic Models
31
at the classical level, the Nambu–Goto-like action (8.1) can be translated into a Polyakov-like
action with an auxiliary metric
Z
1
S 0 [φ] = d4 x (− det g µν )−1/2 − g µν ∂µ φi ∂ν φj δij − F −1 (φ) ,
2
where we are using the auxiliary field g µν .
Notice that this auxiliary field appears algebraically, and hence plays the role of a Lagrange
multiplier. The equations of motion derived from the variation of this classical action lead to
gµν = ∂µ φi ∂ν φj δij F (φ).
Using this equation to eliminate g µν from S 0 leads to the action S.
Clearly, this mapping holds at the classical level. A full quantum theory will presumably
forbid this nice reparametrization of the model. The general idea is that, in the semiclassical
limit of the model, the metric tensor arises as the vev of a composite field
gµν = h∂µ φi ∂ν φj δij F (φ)i.
A semiclassical analysis, following Sakharov’s idea, leads to the conclusion that the dynamic
of this field will be generated through radiative corrections. In particular, the divergent part of
the one loop effective action contains the Lagrangian density
4
N √
Λ
Λ2
Ldiv =
−g
+
R(g) ,
(4π)2
8
24
where N is the number of scalars, R(g) is the Ricci scalar derived from the metric g and Λ is
an energy scale associated to a cutoff to be introduced to regulate the theory.
The scalar model presents a number of issues, in its foundations, that suggest that it should
be taken as a provisional model. The very definition of a path integral for such a theory, notion
of stability, the nondegeneracy of the metric etc are not addressed at all.
Part of these concerns can be mitigated if fermions are used instead of bosons. If fermions
are included, the starting point is the action
Z
i i a
4
j
S[ψ, ψ̄] = d x
ψ̄ γ ∂µ ψ δij F (ψ, ψ̄),
(8.2)
2
where we are introducing Dirac’s gamma matrices γ a . Notice that these last indices are Lorentz/internal indices, and not vector ones, and refer to the fact that the fermions transform
nontrivially under a global Lorentz transformation acting on these indices. Notice also that
there are additional internal indices i, j = 1, . . . , N which manifests the fact that, in order to
have a nonzero result for a partition function, several fermionic fields are required.
In such a model, besides the manifold, one has to postulate the existence of a spin structure.
This is not yet the specification of a metric, given that the definition of a metric tensor still
requires the vielbein. In other terms, in a model with fermions one is forced to introduce a notion
of internal metric, acting on internal/Lorentz indices. A deeper analysis of the situation can be
found in [179].
Given the Grassmannian nature of the fermionic fields, the definition of the quantum partition function is less problematic than the corresponding bosonic case. To extract the content
of the model it is possible to perform a semiclassical analysis. At a semiclassical level, the
action (8.2) is equivalent to the following action
S 0 [ψ, ψ̄] = det(eaµ )−1 eµa ψ̄ i γ a ∂µ ψ j − 3F −1/2 .
32
L. Sindoni
In analogy with the scalar case, the vielbein arises as the expectation value of a composite field,
a fermion bilinear
i
eaµ = hψ̄ i γ a ∂µ ψ j F 1/3 i.
2
Interestingly enough, the action (8.2) is not invariant under local Lorentz transformations, since,
to do so, one should introduce a spin connection, which is of course forbidden by the rules of
emergent gravity. The way in which spin connections, and Yang–Mills gauge fields are introduced
in this model is through the potential term F . Shaping it in such a way to mimick the four
Fermion interaction terms used in models á la Bjorken, we can generate spin connection and
Yang–Mills gauge fields as composite fields.
Indeed, defining
2
−3F −1/3 = α ψ̄ i γ a , [γ b , γ c ] ψ i ,
then the model, at the semiclassical level, becomes equivalent to a local Lorentz invariant model
where the spin connection is determined in terms of the composite field
Aabc = 16α ψ̄ i γ a , [γ b , γ c ] ψ i .
The application of heat kernel techniques to the computation of the one loop effective action
has to take into account the fact that the background is a metric affine one. This generates
radiative corrections that contain, besides the Einstein–Hilbert and the cosmological constant
term, a potential term for the torsion. However, the torsion tensor appears without a kinetic
term, and hence it is not propagating. For further discussions, see [211]. Similar ideas can be
used to define Yang–Mills fields and their dynamics.
Summarizing, these are emergent gravity models, which are locally Lorentz, gauge and
diffeomorphism invariant. This is achieved by the combination of ideas of composite gauge
bosons, semiclassical analysis and quantum field theory that we started from.
These ideas have been developed in various directions [12, 13, 79, 80, 81, 82, 83, 244]. In
[12, 13] it has been presented a model that addresses the spontaneous generation of a mass
scale (crucial to separate UV and IR degrees of freedom) and, simultaneously, the possibility of
unification of Yang–Mills and gravity at high energy. More recently, Wetterich [124, 241, 242,
243] has proposed to use fermionic models to generate gravity along the same lines.
These models are very promising. However, they have to face serious difficulties. Let us
briefly mention them. First of all, one might be worried that a gauge invariant theory might lead
to nonvanishing vacuum expectation values of gauge variant operators, as dictated by Elitzur’s
theorem [93]. However, this difficulty is only in the language, and not in the substance. Indeed,
the presence of a nontrivial vev of a metric tensor or of the vielbein might be expressed in gauge
invariant terms by the equivalent statement that gauge invariant geometrical quantities (areas,
length, angles, volumes) do acquire nonvanishing vev s. The logic is the same behind the gauge
invariant formulation of spontaneous symmetry breaking of gauge field theories with a Higgs
mechanism [169, 208].
A more urgent problem is the very definition of the path integrals. However, while for bosons
the model might seem to be too contrived, for fermions the situation improves considerably and
one might imagine that an appropriate definition might be provided, once the divergences are
appropriately tamed.
A serious problem concerns the conditions under which nonvanishing vev s for geometrical
operators can be generated. In QFT in ordinary Minkowski spacetime, this is controlled by
the specification of the potential of the theory, that in turns determines the properties of the
ground state. The ground state is defined as the state that minimizes the energy, which, in
Emergent Models for Gravity: an Overview of Microscopic Models
33
turn, is defined to be the charge conjugated to a globally defined timelike Killing vector field.
In pregeometry there is no such a thing as a metric: how is the ground state defined, then? How
can we make statements like “at low energy the effective field theory is X”, if we do not have
a notion of energy? In this sense, the emergence of scales has to be addressed, and, after this
issue has been settled, one can imagine to study the RG flow of the theory with the methods
that are being developed in these years, especially in the context of asymptotic safety (see the
contribution in this special issue and also [180] for a discussion of the relevance for the emergent
gravity approach).
However, the most serious problem is the fact that, for these semiclassical arguments to work,
the composite metric or the composite vielbein is assumed to be invertible. This is difficult to
explain from a dynamical perspective (see [179] for a discussion of this point). It might be
that the ultimate reason is some form of Coleman–Weinberg effect in quantum gravity [98],
in which the relevant scales and a potential leading to a nondegenerate metric as a ground
state configuration are generated dynamically by radiative corrections. In absence of such
a dynamical explanation, the only option is to continue to work in a kind of self-consistent mean
field approximation [97, 179].
It seems that the subject is not sufficiently explored. There is a lot of potential for phenomenology given the close contact with the language and the tools of quantum field theories,
for which a vast number of techniques are available. In addition, there is an obvious relevance
for fundamental physics since this class of models can be linked very easily to the issues of
unification of all the interactions. Finally, it seems natural that, in such a scenario, the cosmological singularities of general relativity can be replaced by geometrogenesis phase transitions,
as it is been conjectured on the ground of the experience with quantum graphity models.
9
Lorentz invariance reloaded: emergent signature
Lorentz symmetry plays an important role, in gravitational theory, both at the kinematical
and at the dynamical level. However, in an emergent scenario, it is not obvious that Lorentz
symmetry will be implemented at the fundamental level, as we have seen.
The situation might be that Lorentz invariance is an infrared symmetry, with the ultraviolet
theory being essentially different (see, for instance, anisotropic scaling scenarios [128]). The
situation might be in fact much more drastic. Indeed, the very notion of signature might be
only an emergent one, with the background being more similar to a timeless Euclidean theory
rather than Lorentzian.
Scenarios might be conceived in which the internal Lorentzian metric ηab , in a first order
formalism in which the metric tensor is derived from tetrads gµν = eaµ ebν ηab , is just a specific
vev of another field (a transforming as a tensor in the internal indices). This is the logic behind
the models in which gravity is formulated in terms of GL(4) gauge theories [179]. In this
picture, the selection of Lorentzian rather than Euclidean signatures is a dynamical problem to
be addressed in terms of spontaneous symmetry breaking.
Notice that this is conceptually very different from Wick rotation. A Wick rotation is a procedure of analytic continuation of a theory from Lorentzian to Euclidean signature by analytically
continue the time coordinate to the complex plane, and hence restricting it to the imaginary
axis. The case for dynamical signature is completely different: no analytic continuation is performed. Rather, the theory allows for Lorentzian, Euclidean or more general signatures, at the
level of the equations of motion. The signature is then selected dynamically.
In pregeometrical models like those described in the previous section the situation is similar [243]. The theory is typically formulated in terms of field theories with a large global
symmetry group, and then the signature is selected by the particular symmetry breaking pattern
that accompanies the formation of metrics or tetrads as composite fields.
34
L. Sindoni
This is not the only point of view. In [115] it has been proposed to parametrize the internal
metric as
ηab = diag(eiθ , +1, +1, . . . , +1),
which then smoothly interpolates between the Riemannian and the Lorentzian signatures, keeping the metric nondegenerate (but complex, in general) for any value of θ. This parameter can
inherit its own dynamics from one loop corrections generated by matter fields. It is possible to
compute a form of (complex) effective potential which offers the possibility of computing the
expectation value for the fields itself.
By analyzing the contribution of fermionic and bosonic fields, it is possible to show that
the model can be made consistent only in four spacetime dimensions, by adjusting the matter
content. In turns, the analysis of the potential also predicts Lorentzian signature. Similar ideas
concerning the dynamical generation of signature can be found in [62, 63, 94, 123].
There is a third way to start from a theory which is fundamentally Euclidean but that
manifests Lorentzian effective metrics (and hence propagation of signals). The idea was put
forward in [108] and it is based on the observation that, in a second order PDE, the metric tensor
is identified with (the inverse of) the matrix which is multiplying the second order derivatives:
hµν ∂µ ∂ν φ + aν ∂ν φ + f (φ) = 0.
The signature of the tensor hµν determines whether the PDE is hyperbolic, parabolic or elliptic.
In wave equations encountered normally in physics, the tensor hµν is an externally determined
tensor, independent from the field that is propagating over it. However, when considering
(scalar) field theories which are characterized by (first order!) nonquadratic kinetic terms, the
situation is more complicated. In particular, in a system with Lagrangian
L = F (X),
X = γ µν ∂µ φ∂ν φ,
with γµν a nondegenerate background metric, the principal symbol of the equation of motion is
given by
hµν =
∂F µν
∂ 2 F µρ
γ +2
γ ∂ρ φγ νσ ∂σ φ.
∂X
∂X 2
This tensor shows an explicit dependence on the derivatives of φ. In turn, this can lead to the
presence of solutions for which the tensor hµν is Lorentzian even when γµν is Euclidean. We refer
to [108] for the details of how this can be used to generate a diffeo-invariant emergent Nordström
theory of gravity in a Euclidean flat background. See also [197] for a careful discussion of the
role of the various symmetries that are needed to achieve the result.
Touching a foundational notion of physics as time does not come without a huge price to be
payed. First of all, in models in which Lorentz invariance is emerging from a Euclidean theory
as the last models described, Lorentz symmetry breaking is unavoidable. In models in which
the fundamental symmetry group is larger, like GL(4), or in pregeometrical models described
in [243], the selection of a specific signature via spontaneous symmetry breaking leads to the
appearance of a Nambu–Goldstone mode for each of the generators associated to the broken
symmetries. Approaches as the ones proposed by Greensite are associated to the appearance of
complex actions: there the status of unitarity has to be kept under control.
These problems are just the tip of the iceberg. Indeed, without a definition of Lorentzian
metric, we cannot give a definition of energy and hence of ground state. The classification of
states in terms of energy loses its meaning, and, as a consequence, the selection of states in terms
of energetic considerations (stability, essentially) is not well defined. Furthermore, while any
Emergent Models for Gravity: an Overview of Microscopic Models
35
locally paracompact differentiable manifold can be endowed with a Riemannian structure, it is
not true that we can put a Lorentzian structure over it. For this to be the case, the manifold
should admit the existence of a globally defined, nowhere vanishing smooth vector field. This
is a rather strong topological restriction, which refers to the pregeometric structure.
The emergence of time seems to be a rather exotic curiosity, leading to all sorts of paradoxes.
However, it is a possibility that has to be considered very carefully, since the simplest low energy
theory at our disposal to describe gravity, general relativity, does possess a dynamical content
that seems to be intimately related to the presence of light cones, i.e. to the Lorentzian signature
of the metric tensor. This is clearly purported by the derivations of Einstein equations by means
of thermodynamical considerations on local horizons.
10
Challenges and future directions
Hopefully this overview has given at least the flavor of various attempts to describe gravity as
an emergent dynamical phenomenon, their merits as well as the problems that they raise. The
proposals cover a rather wide range of different points of view and meet different amount of
success and different kinds of shortcomings. We have not mentioned other possibilities, most
notably a specific kind of matrix models for higher dimensional gravity (see [207] and references
therein), matrix models for 2D quantum gravity [85, 107] and their extensions in term of tensor
models [121] and group field theories [99, 170, 172, 173]. These models are extremely interesting
given that their nature is essentially algebraic or combinatoric, and the notions of continuous
differential geometries will emerge in certain specific regimes.
In a certain sense, also the AdS/CFT correspondence is an example of emergent gravity if
seen from the CFT side. However, while on these topics it is easy to find extensive coverage of
the literature, there are many other attempts, that we have tried to present in a way consistent
with the increasing depth at which they attack the problem.
The relevance of this kind of research for quantum gravity might be questioned. However,
it is a fact that in most approaches to quantum gravity the reconstruction of the continuum
macroscopic limit from the underlying microscopic degrees of freedom is a procedure that requires
concepts and tools that are common in statistical models14 and in emergent models. In this sense,
we have tried to give a panoramic view also of the different technical tools that might be useful
to tackle the problem of the continuum limit of quantum gravity models.
As we have seen, the problem of emerging gravity from some pregeometric nongravitational
theory is not yet solved, given that no completely satisfactory model has been worked out.
However, the difficulties encountered do offer also deep insights that are otherwise hardly appreciated in more conventional treatments. Concretely, if gravity is not emergent, but comes
from the semiclassical limit of a theory like loop quantum gravity, still the emergent gravity
models might offer insights and techniques to prove such an assertion.
In closing, we would like to address three final themes.
First of all, it is rather urgent to establish as soon as possible a basis of phenomenological
consequences of ideas of emergent gravity, possibly not in the form of Lorentz violation, on which
an extensive literature on theoretical models and experimental or observational results exists.
A top-down approach is philosophically satisfactory, but cannot exist without confrontation with
experimental evidences from which bottom-up strategies can be elaborated.
There is a rather large body of work on Quantum Gravity phenomenology, that we have
already mentioned. Much of this work is associated to the possibility that the symmetries that
we observe are just low energy concepts, and they might work differently, or being broken, at
high energies. Lorentz invariance has of course a special status, in this respect. However, there
14
On this, see the recent application of renormalization group ideas to spinfoam models [87].
36
L. Sindoni
is much more to do. As we have seen, it is conceivable that diffeomorphism invariance is a low
energy symmetry too, as we have discussed in different places. However, being a local gauge
symmetry (which enters heavily in the determination of the dynamics in theories like GR),
the recovery of a diffeomorphism invariant regime requires that the extra degrees of freedom
decouple at low energy, or that the nondynamical background structures become undetectable.
The investigation of the potential phenomenology associated to diffeo-breaking is a relatively
unexplored territory (see [21]), but it represents an important opportunity to understand how
fundamental diffeomorphism invariance can be considered, and, in the case in which it turned
out to be an approximate symmetry, what is the energy scale that governs the breaking and
what are the associated physical phenomena. As in the case of LIV, this will require, most
significantly, the explanation of the (dynamical) origin of the separation of scales between the
high energy noninvariant phase and the low energy invariant one, to avoid a naturalness problem.
Indeed, in [108] this scale turns out to be the Planck scale, but more general situations might
be conceived.
Another possibility is represented by nonlocality, which is a feature that has been conjectured
long ago to be underlying certain models in which spacetime and hence locality emerges only on
large scales. This is corroborated by many other approaches in which the notion of spacetime
arises either as a structure for the propagation of collective modes as phonons in BEC [199],
or because it arises as a statistical concept as in random models [53, 99, 145], or because
the very topology (in the sense of connectivity) of spacetime looks different at different scales
[163, 186, 205]. See also [104, 105, 106] for a discussion of the role of nonlocality in addressing
some of the puzzles of semiclassical gravity.
Of course, this is only one possibility among the various possible deviations from the rules of
QFT. In general, in pregeometric models, but also in spinfoams and group field theories, there is
one common theme that is the way in which local effective quantum field theories that describe
the theory in the infrared limit are recovered. This is by no means an easy task, and represents
the second theme that is underlying the discussion of emergent models.
Notice also that the fact that, of the various hierarchy problems that plague our effective field
theories, the cosmological constant problem seems to be less severe in an emergent gravity setting
(where the computation of the cosmological constant term is done in term of pregeometrical
degrees of freedom) strongly suggests to further investigate the transition between the effective
field theory regime and the more fundamental layer.
In pursuing this goal, power counting arguments might be of little help. For instance, the
argument that Einstein–Hilbert action terms will be dominating the IR action is not guaranteed
to be valid, given that it is not automatic that the models will contain such operators in their
theory space15 . For this reasons, models with preferred time, or with certain forms of Lorentz
violation might be less compelling than models which, being designed to contain in some form
the Regge action for discretized gravity, might encounter more success.
Of course, this is not still a proof that models like group field theories will deliver the promised
result, given that there is the hard problem of the definition of the continuum limit to be solved.
In addition, in pregeometric scenario the very concept of the renormalization group which gives
theoretical support to the idea of effective field theories has to be reconsidered, in light of the
absence of structures with which we can classify modes in terms of IR and UV. Again, for this
purposes, the tools and ideas (especially about the generation of the dynamical scales) that have
been developed in emergent gravity models might be valuable.
The third important theme is the continuous development of tools from areas like condensed
matter and statistical systems, that will be key ingredients for many (non emergent) quantum
gravity models in the continuum macroscopic limit16 .
15
16
I would like to thank S. Carlip for having raised this point.
On this, see the preliminary results of [174] and [198].
Emergent Models for Gravity: an Overview of Microscopic Models
37
In this sense, the recent use of cold atoms in optical lattices [49, 157] in order to reproduce
effects of quantum fields in extreme configurations does represent a promising new direction.
It has been shown that, in these systems (which are close to the ones underlying, for the constructive principles, quantum graphity models in certain regimes), many interesting phenomena
do occur: the appearance of Dirac operators, abelian and nonabelian artificial fields as well
as the possibility of considering phenomena proper of quantum field theories in extremely intense external field [209, 210] or in which mean field techniques break down [190]. These might
provide a valuable set of hints for the identification of specific effects to look for and for the
some potential difficulties that we might encounter when abandoning the safe harbor of effective
field theories to enter the dangerous waters of models in which the notion of smooth background
structures might have to be revised.
Acknowledgements
I would like to thank A. Campoleoni and F. Caravelli for discussions, and S. Finazzi, F. Girelli,
S. Liberati and D. Oriti for the intense collaboration on the subject. I would also like to thank
T. Gherghetta, S. Hossenfelder, H. Lin and an anonymous referee for comments, constructive
criticism and concrete suggestions for improvements.
References
[1] Adler R., Bazin M., Schiffer M., Introduction to general relativity, McGraw-Hill Book Co., New York, 1965.
[2] Adler S.L., A formula for the induced gravitational constant, Phys. Lett. B 95 (1980), 241–243.
[3] Adler S.L., Einstein gravity as a symmetry-breaking effect in quantum field theory, Rev. Modern Phys. 54
(1982), 729–766, Erratum, Rev. Modern Phys. 55 (1983), 837.
[4] Aharony O., Gubser S.S., Maldacena J., Ooguri H., Oz Y., Large N field theories, string theory and gravity,
Phys. Rep. 323 (2000), 183–386, hep-th/9905111.
[5] Akama K., An attempt at pregeometry. Gravity with composite metric, Progr. Theoret. Phys. 60 (1978),
1900–1909.
[6] Akama K., Pregeometry including fundamental gauge bosons, Phys. Rev. D 24 (1981), 3073–3081.
[7] Akama K., Chikashige Y., Matsuki T., Unified model of the Nambu–Jona–Lasinio type for the gravitational
and electromagnetic forces, Progr. Theoret. Phys. 59 (1978), 653–655.
[8] Akama K., Chikashige Y., Matsuki T., Terazawa H., Gravity and electromagnetism as collective phenomena
of fermion-antifermion pairs, Progr. Theoret. Phys. 60 (1978), 868–877.
[9] Alexander S.H.S., Calcagni G., Quantum gravity as a Fermi liquid, Found. Phys. 38 (2008), 1148–1184,
arXiv:0807.0225.
[10] Alexander S.H.S., Calcagni G., Superconducting loop quantum gravity and the cosmological constant, Phys.
Lett. B 672 (2009), 386–389, arXiv:0806.4382.
[11] Alfaro J., Morales-Técotl H.A., Urrutia L.F., Quantum gravity corrections to neutrino propagation, Phys.
Rev. Lett. 84 (2000), 2318–2321, gr-qc/9909079.
[12] Amati D., Veneziano G., A unified gauge and gravity theory with only matter fields, Nuclear Phys. B 204
(1981), 451–476.
[13] Amati D., Veneziano G., Metric from matter, Phys. Lett. B 105 (1981), 358–362.
[14] Ambjørn J., Görlich A., Jurkiewicz J., Loll R., CDT – an entropic theory of quantum gravity,
arXiv:1007.2560.
[15] Ambjørn J., Jurkiewicz J., Loll R., Causal dynamical triangulations and the quest for quantum gravity,
arXiv:1004.0352.
[16] Ambjørn J., Jurkiewicz J., Loll R., Lattice quantum gravity – an update, PoS Proc. Sci. (2010),
PoS(LATTICE2010), 014, 14 pages, arXiv:1105.5582.
[17] Ambjørn J., Jurkiewicz J., Loll R., Quantum gravity as sum over spacetimes, in New Paths Towards
Quantum Gravity, Lecture Notes in Physics, Vol. 807, Springer, Berlin, 2010, 59–124, arXiv:0906.3947.
38
L. Sindoni
[18] Ambjørn J., Jurkiewicz J., Loll R., Quantum gravity, or the art of building spacetime, in Approaches to
Quantum Gravity, Editor D. Oriti, Cambridge University Press, Cambridge, 2009, 341–359, hep-th/0604212.
[19] Amelino-Camelia G., Are we at the dawn of quantum-gravity phenomenology?, in Towards Quantum Gravity
(Polanica, 1999), Lecture Notes in Phys., Vol. 541, Springer, Berlin, 2000, 1–49, gr-qc/9910089.
[20] Amelino-Camelia G., Quantum gravity phenomenology, arXiv:0806.0339.
[21] Anber M.M., Aydemir U., Donoghue J.F., Breaking diffeomorphism invariance and tests for the emergence
of gravity, Phys. Rev. D 81 (2010), 084059, 12 pages, arXiv:0911.4123.
[22] Atkatz D., Dynamical method for generating the gravitational interaction, Phys. Rev. D 17 (1978), 1972–
1976.
[23] Balasubramanian V., de Boer J., Jejjala V., Simón J., The library of Babel: on the origin of gravitational
thermodynamics, J. High Energy Phys. 2005 (2005), no. 12, 006, 65 pages, hep-th/0508023.
[24] Balbinot R., Fagnocchi S., Fabbri A., Quantum effects in acoustic black holes: the backreaction, Phys.
Rev. D 71 (2005), 064019, 12 pages, gr-qc/0405098.
[25] Balbinot R., Fagnocchi S., Fabbri A., Procopio G.P., Backreaction in acoustic black holes, Phys. Rev. Lett.
94 (2005), 161302, 4 pages, gr-qc/0405096.
[26] Banks T., Zaks A., Composite gauge bosons in 4-fermi theories, Nuclear Phys. B 184 (1981), 303–322.
[27] Bao D., Chern S.S., Shen Z., An introduction to Riemann–Finsler geometry, Graduate Texts in Mathematics,
Vol. 200, Springer-Verlag, New York, 2000.
[28] Barbado L.C., Barceló C., Garay L.J., Jannes G., The trans-Planckian problem as a guiding principle,
J. High Energy Phys. 2011 (2011), no. 11, 112, 18 pages, arXiv:1109.3593.
[29] Barceló C., Garay L.J., Jannes G., Sensitivity of Hawking radiation to superluminal dispersion relations,
Phys. Rev. D 79 (2009), 024016, 13 pages, arXiv:0807.4147.
[30] Barceló C., Garay L.J., Jannes G., Two faces of quantum sound, Phys. Rev. D 82 (2010), 044042, 12 pages,
arXiv:1006.0181.
[31] Barceló C., Jannes G., A real Lorentz–FitzGerald contraction, Found. Phys. 38 (2008), 191–199,
arXiv:0705.4652.
[32] Barceló C., Liberati S., Visser M., Analogue gravity, Living Rev. Relativ. 14 (2011), 3, 159 pages,
gr-qc/0505065.
[33] Barceló C., Liberati S., Visser M., Analogue gravity from Bose–Einstein condensates, Classical Quantum
Gravity 18 (2001), 1137–1156, gr-qc/0011026.
[34] Barceló C., Liberati S., Visser M., Analogue models for FRW cosmologies, Internat. J. Modern Phys. D 12
(2003), 1641–1649, gr-qc/0305061.
[35] Barceló C., Liberati S., Visser M., Refringence, field theory and normal modes, Classical Quantum Gravity
19 (2002), 2961–2982, gr-qc/0111059.
[36] Barceló C., Visser M., Liberati S., Einstein gravity as an emergent phenomenon?, Internat. J. Modern
Phys. D 10 (2001), 799–806, gr-qc/0106002.
[37] Bardeen J.M., Carter B., Hawking S.W., The four laws of black hole mechanics, Comm. Math. Phys. 31
(1973), 161–170.
[38] Bardeen J.M., Cooper L.N., Schrieffer J.R., Theory of superconductivity, Phys. Rev. 108 (1957), 1175–1204.
[39] Bekaert X., Boulanger N., Sundell P., How higher-spin gravity surpasses the spin two barrier: no-go theorems
versus yes-go examples, arXiv:1007.0435.
[40] Bekenstein J.D., Black holes and entropy, Phys. Rev. D 7 (1973), 2333–2346.
[41] Belgiorno F., Cacciatori S.L., Clerici M., Gorini V., Ortenzi G., Rizzi L., Rubino E., Sala V.G., Hawking radiation from ultrashort laser pulse filaments, Phys. Rev. Lett. 105 (2010), 203901, 4 pages, arXiv:1009.4634.
[42] Benedetti D., Henson J., Imposing causality on a matrix model, Phys. Lett. B 678 (2009), 222–226,
arXiv:0812.4261.
[43] Berenstein D., Large N BPS states and emergent quantum gravity, J. High Energy Phys. 2006 (2006), no. 1,
125, 58 pages, hep-th/0507203.
[44] Berezhiani Z., Kancheli O.V., Spontaneous breaking of Lorentz-invariance and gravitons as Goldstone particles, arXiv:0808.3181.
[45] Bettoni D., Liberati S., Sindoni L., Extended ΛCDM: generalized non-minimal coupling for dark matter
fluids, J. Cosmol. Astropart. Phys. 2011 (2011), no. 11, 007, 17 pages, arXiv:1108.1728.
Emergent Models for Gravity: an Overview of Microscopic Models
39
[46] Bialynicki-Birula I., Quantum electrodynamics without electromagnetic field, Phys. Rev. 130 (1963), 465–
468.
[47] Bjorken J.D., A dynamical origin for the electromagnetic field, Ann. Physics 24 (1963), 174–187.
[48] Bjorken J.D., Emergent gauge bosons, hep-th/0111196.
[49] Bloch I., Dalibard J., Zwerger W., Many-body physics with ultracold gases, Rev. Modern Phys. 80 (2008),
885–964, arXiv:0704.3011.
[50] Bluhm R., Fung S.H., Kostelecký V.A., Spontaneous Lorentz and diffeomorphism violation, massive modes,
and gravity, Phys. Rev. D 77 (2008), 065020, 28 pages, arXiv:0712.4119.
[51] Bluhm R., Kostelecký V.A., Spontaneous Lorentz violation, Nambu–Goldstone modes, and gravity, Phys.
Rev. D 71 (2005), 065008, 17 pages, hep-th/0412320.
[52] Bombelli L., Henson J., Sorkin R.D., Discreteness without symmetry breaking: a theorem, Modern Phys.
Lett. A 24 (2009), 2579–2587, gr-qc/0605006.
[53] Bombelli L., Lee J., Meyer D., Sorkin R.D., Space-time as a causal set, Phys. Rev. Lett. 59 (1987), 521–524.
[54] Born M., Wolf E., Principles of optics, Cambridge University Press, Cambridge, 1999.
[55] Brout R., Massar S., Parentani R., Spindel P., Hawking radiation without trans-Planckian frequencies, Phys.
Rev. D 52 (1995), 4559–4568, hep-th/9506121.
[56] Buchbinder I.L., Odintsov S.D., Shapiro I.L., Effective action in quantum gravity, IOP Publishing Ltd.,
Bristol, 1992.
[57] Burgess C.P., Goldstone and pseudo-Goldstone bosons in nuclear, particle and condensed-matter physics,
Phys. Rep. 330 (2000), 193–261, hep-th/9808176.
[58] Cadoni M., Mignemi S., Acoustic analogues of black hole singularities, Phys. Rev. D 72 (2005), 084012,
9 pages, gr-qc/0504143.
[59] Calzetta E.A., Hu B.L., Early universe quantum processes in BEC Collapse Experiments, Internat. J.
Theoret. Phys. 44 (2005), 1691–1704, cond-mat/0503367.
[60] Caravelli F., Hamma A., Markopoulou F., Riera A., Trapped surfaces and emergent curved space in the
Bose–Hubbard model, Phys. Rev. D 85 (2012), 044046, 15 pages, arXiv:1108.2013.
[61] Caravelli F., Markopoulou F., Properties of quantum graphity at low temperature, Phys. Rev. D 84 (2011),
024002, 10 pages, arXiv:1008.1340.
[62] Carlini A., Greensite J., Square-root actions, metric signature, and the path integral of quantum gravity,
Phys. Rev. D 52 (1995), 6947–6964, gr-qc/9502023.
[63] Carlini A., Greensite J., Why is spacetime Lorentzian?, Phys. Rev. D 49 (1994), 866–878, gr-qc/9308012.
[64] Carroll S.M., The cosmological constant, Living Rev. Relativ. 4 (2001), 1, 80 pages, astro-ph/0004075.
[65] Carroll S.M., Tam H., Wehus I.K., Lorentz violation in Goldstone gravity, Phys. Rev. D 80 (2009), 025020,
15 pages, arXiv:0904.4680.
[66] Case K.M., Gasiorowicz S.G., Can massless particles by charged?, Phys. Rev. 125 (1962), 1055–1058.
[67] Chirco G., Eling C., Liberati S., Reversible and irreversible spacetime thermodynamics for general Brans–
Dicke theories, Phys. Rev. D 83 (2011), 024032, 12 pages, arXiv:1011.1405.
[68] Chirco G., Liberati S., Nonequilibrium thermodynamics of spacetime: the role of gravitational dissipation,
Phys. Rev. D 81 (2010), 024016, 13 pages, arXiv:0909.4194.
[69] Chkareuli J.L., Froggatt C.D., Nielsen H.B., Deriving gauge symmetry and spontaneous Lorentz violation,
Nuclear Phys. B 821 (2009), 65–73, hep-th/0610186.
[70] Chkareuli J.L., Froggatt C.D., Nielsen H.B., Lorentz invariance and origin of symmetries, Phys. Rev. Lett.
87 (2001), 091601, 4 pages, hep-ph/0106036.
[71] Chkareuli J.L., Froggatt C.D., Nielsen H.B., Spontaneously generated gauge invariance, Nuclear Phys. B
609 (2001), 46–60, hep-ph/0103222.
[72] Chkareuli J.L., Froggatt C.D., Nielsen H.B., Spontaneously generated tensor field gravity, Nuclear Phys. B
848 (2011), 498–522, arXiv:1102.5440.
[73] Chkareuli J.L., Jejelava J.G., Tatishvili G., Graviton as a Goldstone boson: nonlinear sigma model for tensor
field gravity, Phys. Lett. B 696 (2011), 124–130, arXiv:1008.3707.
[74] Cho Y.M., Freund P.G.O., Non-Abelian gauge fields as Nambu–Goldstone fields, Phys. Rev. D 12 (1975),
1711–1720.
40
L. Sindoni
[75] Collins J., Perez A., Sudarsky D., Urrutia L., Vucetich H., Lorentz invariance and quantum gravity: an
additional fine-tuning problem?, Phys. Rev. Lett. 93 (2004), 191301, 4 pages, gr-qc/0403053.
[76] Conrady F., Space as a low-temperature regime of graphs, J. Stat. Phys. 142 (2011), 898–917,
arXiv:1009.3195.
[77] Corley S., Jacobson T., Hawking spectrum and high frequency dispersion, Phys. Rev. D 54 (1996), 1568–
1586, hep-th/9601073.
[78] de Mello Koch R., Geometries from Young diagrams, J. High Energy Phys. 2008 (2008), no. 11, 061,
30 pages, arXiv:0806.0685.
[79] Denardo G., Spallucci E., Curvature and symmetry breaking: an induced-action approach, Nuovo Cimento A
71 (1982), 397–408.
[80] Denardo G., Spallucci E., Curvature and torsion from matter, Classical Quantum Gravity 4 (1987), 89–99.
[81] Denardo G., Spallucci E., Finite-temperature scalar pregeometry, Nuovo Cimento A 74 (1983), 450–460.
[82] Denardo G., Spallucci E., Finite temperature spinor pregeometry, Phys. Lett. B 130 (1983), 43–46.
[83] Denardo G., Spallucci E., Induced quantum gravity from heat kernel expansion, Nuovo Cimento A 69
(1982), 151–159.
[84] Deser S., Gravity from self-interaction redux, Gen. Relativity Gravitation 42 (2010), 641–646,
arXiv:0910.2975.
[85] Di Francesco P., Ginsparg P., Zinn-Justin J., 2D gravity and random matrices, Phys. Rep. 254 (1995), 133,
hep-th/9306153.
[86] Distler J., Garibaldi S., There is no “theory of everything” inside E8 , Comm. Math. Phys. 298 (2010),
419–436, arXiv:0905.2658.
[87] Dittrich B., Eckert F.C., Martin-Benito M., Coarse graining methods for spin net and spin foam models,
New J. Phys. 14 (2012), 035008, 43 pages, arXiv:1109.4927.
[88] Dreyer O., Emergent general relativity, in Approaches to Quantum Gravity, Editor D. Oriti, Cambridge
University Press, Cambridge, 2009, 99–110, gr-qc/0604075.
[89] Dreyer O., Why things fall, PoS Proc. Sci. (2007), PoS(QG–Ph), 016, 12 pages, arXiv:0710.4350.
[90] Eguchi T., New approach to collective phenomena in superconductivity models, Phys. Rev. D 14 (1976),
2755–2763.
[91] Eguchi T., Sugawara H., Extended model of elementary particles based on an analogy with superconductivity, Phys. Rev. D 10 (1974), 4257–4262.
[92] Eling C., Guedens R., Jacobson T., Nonequilibrium thermodynamics of spacetime, Phys. Rev. Lett. 96
(2006), 121301, 4 pages, gr-qc/0602001.
[93] Elitzur S., Impossibility of spontaneously breaking local symmetries, Phys. Rev. D 12 (1975), 3978–3982.
[94] Elizalde E., Odintsov S.D., Romeo A., Dynamical determination of the metric signature in spacetime of
nontrivial topology, Classical Quantum Gravity 11 (1994), L61–L67, hep-th/9312132.
[95] Finazzi S., Liberati S., Sindoni L., Cosmological constant: a Lesson from Bose–Einstein condensates, Phys.
Rev. Lett. 108 (2012), 071101, 5 pages, arXiv:1103.4841.
[96] Finazzi S., Parentani R., Spectral properties of acoustic black hole radiation: broadening the horizon, Phys.
Rev. D 83 (2011), 084010, 13 pages, arXiv:1012.1556.
[97] Floreanini R., Percacci R., Mean-field quantum gravity, Phys. Rev. D 46 (1992), 1566–1579.
[98] Floreanini R., Percacci R., Spallucci E., Coleman–Weinberg effect in quantum gravity, Classical Quantum
Gravity 8 (1991), L193–L197.
[99] Freidel L., Group field theory:
hep-th/0505016.
an overview, Internat. J. Theoret. Phys. 44 (2005), 1769–1783,
[100] Gambini R., Pullin J., Nonstandard optics from quantum space-time, Phys. Rev. D 59 (1999), 124021,
4 pages, gr-qc/9809038.
[101] Garay L.J., Anglin J.R., Cirac J.I., Zoller P., Sonic analog of gravitational black holes in Bose–Einstein
condensates, Phys. Rev. Lett. 85 (2000), 4643–4647, gr-qc/0002015.
[102] Garay L.J., Anglin J.R., Cirac J.I., Zoller P., Sonic black holes in dilute Bose–Einstein condensates, Phys.
Rev. A 63 (2001), 023611, 13 pages, gr-qc/0005131.
[103] Gherghetta T., Peloso M., Poppitz E., Emergent gravity from a mass deformation in warped spacetime,
Phys. Rev. D 72 (2005), 104003, 23 pages, hep-th/0507245.
Emergent Models for Gravity: an Overview of Microscopic Models
41
[104] Giddings S.B., Black hole information, unitarity, and nonlocality, Phys. Rev. D 74 (2006), 106005, 15 pages,
hep-th/0605196.
[105] Giddings S.B., (Non)perturbative gravity, nonlocality, and nice slices, Phys. Rev. D 74 (2006), 106009,
10 pages, hep-th/0606146.
[106] Giddings S.B., Lippert M., The information paradox and the locality bound, Phys. Rev. D 69 (2004),
124019, 9 pages, hep-th/0402073.
[107] Ginsparg P., Matrix models of 2d gravity, hep-th/9112013.
[108] Girelli F., Liberati S., Sindoni L., Emergence of Lorentzian signature and scalar gravity, Phys. Rev. D 79
(2009), 044019, 9 pages, arXiv:0806.4239.
[109] Girelli F., Liberati S., Sindoni L., Gravitational dynamics in Bose–Einstein condensates, Phys. Rev. D 78
(2008), 084013, 11 pages, arXiv:0807.4910.
[110] Girelli F., Liberati S., Sindoni L., Planck-scale modified dispersion relations and Finsler geometry, Phys.
Rev. D 75 (2007), 064015, 9 pages, gr-qc/0611024.
[111] Girelli F., Livine E.R., A deformed Poincaré invariance for group field theories, Classical Quantum Gravity
27 (2010), 245018, 15 pages, arXiv:1001.2919.
[112] Girelli F., Livine E.R., Oriti D., Deformed special relativity as an effective flat limit of quantum gravity,
Nuclear Phys. B 708 (2005), 411–433, gr-qc/0406100.
[113] Girelli F., Livine E.R., Oriti D., Four-dimensional deformed special relativity from group field theories,
Phys. Rev. D 81 (2010), 024015, 14 pages, arXiv:0903.3475.
[114] Giulini D., Remarks on the notions of general covariance and background independence, in Approaches to
Fundamental Physics, Lecture Notes in Phys., Vol. 721, Springer, Berlin, 2007, 105–120, gr-qc/0603087.
[115] Greensite J., Dynamical origin of the Lorentzian signature of spacetime, Phys. Lett. B 300 (1993), 34–37,
gr-qc/9210008.
[116] Groot Nibbelink S., Pospelov M., Lorentz violation in supersymmetric field theories, Phys. Rev. Lett. 94
(2005), 081601, 4 pages, hep-ph/0404271.
[117] Gu Z.C., Wen X.G., A lattice bosonic model as a quantum theory of gravity, gr-qc/0606100.
[118] Gu Z.C., Wen X.G., Emergence of helicity ±2 modes (gravitons) from qubit models, arXiv:0907.1203.
[119] Guralnik G.S., Photon as a symmetry-breaking solution to field theory. I, Phys. Rev. 136 (1964), B1404–
B1416.
[120] Guralnik G.S., Photon as a symmetry-breaking solution to field theory. II, Phys. Rev. 136 (1964), B1417–
B1422.
[121] Gurau R., Ryan J.P., Colored tensor models – a review, SIGMA 8 (2012), 020, 78 pages, arXiv:1109.4812.
[122] Hamma A., Markopoulou F., Lloyd S., Caravelli F., Severini S., Markstrom K., Quantum Bose–Hubbard
model with an evolving graph as a toy model for emergent spacetime, Phys. Rev. D 81 (2010), 104032,
22 pages, arXiv:0911.5075.
[123] Hayward S.A., Complex lapse, complex action, and path integrals, Phys. Rev. D 53 (1996), 5664–5669,
gr-qc/9511007.
[124] Hebecker A., Wetterich C., Spinor gravity, Phys. Lett. B 574 (2003), 269–275, hep-th/0307109.
[125] Hehl F.W., Obukhov Y.N., Spacetime metric from local and linear electrodynamics: a new axiomatic
scheme, in Special Relativity, Lecture Notes in Physics, Vol. 702, Springer, Berlin, 2006, 163–187,
gr-qc/0508024.
[126] Henson J., The causal set approach to quantum gravity, in Approaches to Quantum Gravity, Editor D. Oriti,
Cambridge University Press, Cambridge, 2009, 393–413, gr-qc/0601121.
[127] Hojman S.A., Kuchař K., Teitelboim C., Geometrodynamics regained, Ann. Physics 96 (1976), 88–135.
[128] Horava P., Quantum gravity at a Lifshitz point, Phys. Rev. D 79 (2009), 084008, 15 pages, arXiv:0901.3775.
[129] Horava P., Stability of fermi surfaces and K theory, Phys. Rev. Lett. 95 (2005), 016405, 4 pages,
hep-th/0503006.
[130] Horowitz G.T., Polchinski J., Gauge/gravity duality, in Approaches to Quantum Gravity, Editor D. Oriti,
Cambridge University Press, Cambridge, 2009, 169–186, gr-qc/0602037.
[131] Hossenfelder S., Experimental search for quantum gravity, in Classical and Quantum Gravity: Theory, Analysis and Applications, Editor V.R. Frignanni, Nova Publishers, New York, 2011, Chapter 5,
arXiv:1010.3420.
42
L. Sindoni
[132] Hu B.L., Can spacetime be a condensate?, Internat. J. Theoret. Phys. 44 (2005), 1785–1806, gr-qc/0503067.
[133] Hu B.L., Emergent/quantum gravity: macro/micro structures of spacetime, J. Phys. Conf. Ser. 174 (2009),
012015, 16 pages, arXiv:0903.0878.
[134] Hu B.L., General relativity as geometrohydrodynamics, gr-qc/9607070.
[135] Hu B.L., Gravity and nonequilibrium thermodynamics of classical matter, Internat. J. Modern Phys. D 20
(2011), 697–716, arXiv:1010.5837.
[136] Hubeny V.E., Minwalla S., Rangamani M., The fluid/gravity correspondence, arXiv:1107.5780.
[137] Iengo R., Russo J.G., Serone M., Renormalization group in Lifshitz-type theories, J. High Energy Phys.
2009 (2009), no. 11, 020, 25 pages, arXiv:0906.3477.
[138] Jacobson T., Thermodynamics of spacetime: the Einstein equation of state, Phys. Rev. Lett. 75 (1995),
1260–1263, gr-qc/9504004.
[139] Jannes G., Emergent gravity: the BEC paradigm, Ph.D. thesis, Universidad Complutense de Madrid, 2009,
arXiv:0907.2839.
[140] Jannes G., Volovik G.E., The cosmological constant: a lesson from topological Weyl media, arXiv:1108.5086.
[141] Jenkins A., Constraints on emergent gravity, Internat. J. Modern Phys. D 18 (2009), 2249–2255,
arXiv:0904.0453.
[142] Jenkins A., Topics in particle physics and cosmology beyond the standard model, Ph.D. thesis, California
Institute of Technology, 2006, hep-th/0607239.
[143] Konopka T., Statistical mechanics of graphity models, Phys. Rev. D 78 (2008), 044032, 17 pages,
arXiv:0805.2283.
[144] Konopka T., Markopoulou F., Severini S., Quantum graphity: a model of emergent locality, Phys. Rev. D
77 (2008), 104029, 15 pages, arXiv:0801.0861.
[145] Konopka T., Markopoulou F., Smolin L., Quantum graphity, hep-th/0611197.
[146] Kostelecký V.A., Potting R., Gravity from spontaneous Lorentz violation, Phys. Rev. D 79 (2009), 065018,
21 pages, arXiv:0901.0662.
[147] Kostelecký V.A., Tasson J.D., Matter-gravity couplings and Lorentz violation, Phys. Rev. D 83 (2011),
016013, 59 pages, arXiv:1006.4106.
[148] Kraus P., Tomboulis E.T., Photons and gravitons as Goldstone bosons and the cosmological constant, Phys.
Rev. D 66 (2002), 045015, 10 pages, hep-th/0203221.
[149] Lahav O., Itah A., Blumkin A., Gordon C., Steinhauer J., Numerical observation of Hawking radiation
from acoustic black holes in atomic Bose–Einstein condensates, New J. Phys. 10 (2008), 103001, 15 pages,
arXiv:0803.0507.
[150] Lahav O., Itah A., Blumkin A., Gordon C., Steinhauer J., Realization of a sonic black hole analog in
a Bose–Einstein condensate, Phys. Rev. Lett. 105 (2010), 240401, 4 pages, arXiv:0906.1337.
[151] Landau L.D., Lifshitz E.M., Course of theoretical physics, Vol. 7. Theory of elasticity, Pergamon Press,
London, 1959.
[152] Landau L.D., Lifshitz E.M., Course of theoretical physics, Vol. 8. Electrodynamics of continuous media,
Pergamon Press, Oxford, 1960.
[153] Laugwitz D., Differential and Riemannian geometry, Academic Press, New York, 1965.
[154] Levin M., Wen X.G., Colloquium: Photons and electrons as emergent phenomena, Rev. Modern Phys. 77
(2005), 871–879, cond-mat/0407140.
[155] Levin M., Wen X.G., Fermions, strings, and gauge fields in lattice spin models, Phys. Rev. B 67 (2003),
245316, 10 pages, cond-mat/0302460.
[156] Levin M., Wen X.G., Quantum ether: photons and electrons from a rotor model, Phys. Rev. B 73 (2006),
035122, 10 pages, hep-th/0507118.
[157] Lewenstein M., Sanpera A., Ahufinger V., Damski B., Sen De A., Sen U., Ultracold atomic gases
in optical lattices: mimicking condensed matter physics and beyond, Adv. Phys. 56 (2007), 243–379,
cond-mat/0606771.
[158] Liberati S., Maccione L., Quantum gravity phenomenology: achievements and challenges, J. Phys. Conf.
Ser. 314 (2011), 012007, 10 pages, arXiv:1105.6234.
[159] Liberati S., Sindoni L., Sonego S., Linking the trans-Planckian and information loss problems in black hole
physics, Gen. Relativity Gravitation 42 (2010), 1139–1152, arXiv:0904.0815.
Emergent Models for Gravity: an Overview of Microscopic Models
43
[160] Liberati S., Visser M., Weinfurtner S., Analogue quantum gravity phenomenology from a two-component
Bose–Einstein condensate, Classical Quantum Gravity 23 (2006), 3129–3154, gr-qc/0510125.
[161] Lin H., Lunin O., Maldacena J., Bubbling AdS space and 1/2 BPS geometries, J. High Energy Phys. 2004
(2004), no. 10, 025, 68 pages, hep-th/0409174.
[162] Loebbert F., The Weinberg–Witten theorem on massless particles: an essay, Ann. Phys. 17 (2008), 803–829.
[163] Markopoulou F., Smolin L., Disordered locality in loop quantum gravity states, Classical Quantum Gravity
24 (2007), 3813–3823, gr-qc/0702044.
[164] Mathur S.D., The information paradox: a pedagogical introduction, Classical Quantum Gravity 26 (2009),
224001, 31 pages, arXiv:0909.1038.
[165] Mattingly D., Have we tested Lorentz invariance enough?, PoS Proc. Sci. (2007), PoS(QG–Ph), 026,
17 pages, arXiv:0802.1561.
[166] Nambu Y., Jona-Lasinio G., Dynamical model of elementary particles based on an analogy with superconductivity. I, Phys. Rev. 122 (1961), 345–358.
[167] Nambu Y., Jona-Lasinio G., Dynamical model of elementary particles based on an analogy with superconductivity. II, Phys. Rev. 124 (1961), 246–254.
[168] Ohanian H.C., Gravitons as Goldstone bosons, Phys. Rev. 184 (1969), 1305–1312.
[169] O’Raifeartaigh L., Hidden gauge symmetry, Rep. Progr. Phys. 42 (1979), 159–223.
[170] Oriti D., Group field theory as the microscopic description of the quantum spacetime fluid: a new
perspective on the continuum in quantum gravity, PoS Proc. Sci. (2007), PoS(QG–Ph), 030, 38 pages,
arXiv:0710.3276.
[171] Oriti D., On the depth of quantum space, arXiv:1107.4534.
[172] Oriti D., The group field theory approach to quantum gravity, in Approaches to Quantum Gravity, Editor
D. Oriti, Cambridge University Press, Cambridge, 2009, 310–331, gr-qc/0607032.
[173] Oriti D., The microscopic dynamics of quantum space as a group field theory, arXiv:1110.5606.
[174] Oriti D., Sindoni L., Towards classical geometrodynamics from group field theory hydrodynamics, New J.
Phys. 13 (2011), 025006, 44 pages, arXiv:1010.5149.
[175] Padmanabhan T., Cosmological constant – the weight of the vacuum, Phys. Rep. 380 (2003), 235–320,
hep-th/0212290.
[176] Padmanabhan T., Gravity and the thermodynamics of horizons, Phys. Rep. 406 (2005), 49–125,
gr-qc/0311036.
[177] Padmanabhan T., Thermodynamical aspects of gravity: new insights, Rep. Progr. Phys. 73 (2010), 046901,
44 pages, arXiv:0911.5004.
[178] Padmanabhan T., Vacuum fluctuations of energy density can lead to the observed cosmological constant,
Classical Quantum Gravity 22 (2005), L107–L112, hep-th/0406060.
[179] Percacci R., The Higgs phenomenon in quantum gravity, Nuclear Phys. B 353 (1991), 271–290,
arXiv:0712.3545.
[180] Percacci R., Vacca G.P., Asymptotic safety, emergence and minimal length, Classical Quantum Gravity 27
(2010), 245026, 16 pages, arXiv:1008.3621.
[181] Perez A., Spin foam models for quantum gravity, Classical Quantum Gravity 20 (2003), R43–R104,
gr-qc/0301113.
[182] Pethick C.J., Smith H., Bose–Einstein condensation in dilute gases, 2nd ed., Cambridge University Press,
Cambridge, 2008.
[183] Phillips P.R., Is the graviton a Goldstone boson?, Phys. Rev. 166 (1964), 966–973.
[184] Polchinski J., String theory. Vol. I. An introduction to the bosonic string, Cambridge Monographs on
Mathematical Physics, Cambridge University Press, Cambridge, 1998.
[185] Prain A., Fagnocchi S., Liberati S., Analogue cosmological particle creation: quantum correlations in expanding Bose–Einstein condensates, Phys. Rev. D 82 (2010), 105018, 18 pages, arXiv:1009.0647.
[186] Prescod-Weinstein C., Smolin L., Disordered locality as an explanation for the dark energy, Phys. Rev. D
80 (2009), 063505, 5 pages, arXiv:0903.5303.
[187] Punzi R., Schuller F.P., Wohlfarth M.N.R., Area metric gravity and accelerating cosmology, J. High Energy
Phys. 2007 (2007), no. 2, 030, 44 pages, hep-th/0612141.
[188] Punzi R., Schuller F.P., Wohlfarth M.N.R., Massive motion in area metric spacetimes, Phys. Rev. D 79
(2009), 124025, 8 pages.
44
L. Sindoni
[189] Rätzel D., Rivera S., Schuller F.P., Geometry of physical dispersion relations, Phys. Rev. D 83 (2011),
044047, 23 pages, arXiv:1010.1369.
[190] Rey A.M., Hu B.L., Calzetta E., Roura A., Clark C.W., Nonequilibrium dynamics of optical-lattice-loaded
Bose–Einstein-condensate atoms: beyond the Hartree–Fock–Bogoliubov approximation, Phys. Rev. A 69
(2004), 033610, 21 pages, cond-mat/0308305.
[191] Rovelli C., Quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press,
Cambridge, 2004.
[192] Rund H., The differential geometry of Finsler spaces, Die Grundlehren der Mathematischen Wissenschaften,
Bd. 101, Springer-Verlag, Berlin, 1959.
[193] Sakharov A.D., Vacuum quantum fluctuations in curved space and the theory of gravitation, Gen. Relativity
Gravitation 32 (2000), 365–367 (Translated from Dokl. Akad. Nauk 1 (1967), no. 1, 70–71).
[194] Schrödinger E., Space-time structure, Cambridge University Press, Cambridge, 1950.
[195] Seiberg N., Emergent spacetime, hep-th/0601234.
[196] Sindoni L., Emergent gravitational dynamics from multi-Bose–Einstein-condensate hydrodynamics?, Phys.
Rev. D 83 (2011), 024022, 18 pages, arXiv:1011.4411.
[197] Sindoni L., Emergent gravity: the analogue models perspective, Ph.D. thesis, SISSA, Trieste, Italy, 2009.
[198] Sindoni L., Gravity as an emergent phenomenon: a GFT perspective, arXiv:1105.5687.
[199] Sindoni L., Girelli F., Liberati S., Emergent gravitational dynamics in Bose–Einstein condensates, AIP
Conf. Proc. 1196 (2009), 258–265, arXiv:0909.5391.
[200] Skákala J., Aspects of general relativity: pseudo-Finsler extensions, quasi-normal frequencies and multiplication of tensorial distributions, Ph.D. thesis, Victoria University of Wellington, 2011, arXiv:1107.2978.
[201] Skákala J., Visser M., Bi-metric pseudo-Finslerian spacetimes, J. Geom. Phys. 61 (2011), 1396–1400,
arXiv:1008.0689.
[202] Skákala J., Visser M., Birefringence in pseudo-Finsler spacetimes, J. Phys. Conf. Ser. 189 (2009), 012037,
8 pages, arXiv:0810.4376.
[203] Skákala J., Visser M., Pseudo-Finslerian space-times and multirefringence, Internat. J. Modern Phys. D 19
(2010), 1119–1146, arXiv:0806.0950.
[204] Skákala J., Visser M., The causal structure of spacetime is a parameterized Randers geometry, Classical
Quantum Gravity 28 (2011), 065007, 7 pages, arXiv:1012.4467.
[205] Sorkin R.D., Does locality fail at intermediate length-scales, in Approaches to Quantum Gravity, Editor
D. Oriti, Cambridge University Press, Cambridge, 2009, 26–43, gr-qc/0703099.
[206] Sorkin R.D., Is the cosmological “constant” a nonlocal quantum residue of discreteness of the causal set
type?, AIP Conf. Proc. 957 (2007), 142–153, arXiv:0710.1675.
[207] Steinacker H., Non-commutative geometry and matrix models, arXiv:1109.5521.
[208] Struyve W., Gauge invariant accounts of the Higgs mechanism, Stud. Hist. Philos. Sci. B Stud. Hist. Philos.
Modern Phys. 42 (2011), 226–236, arXiv:1102.0468.
[209] Szpak N., Schutzhold R., Optical lattice quantum simulator for quantum electrodynamics in strong external
fields: spontaneous pair creation and the Sauter-Schwinger effect, New J. Phys. 14 (2012), 035001, 23 pages,
arXiv:1109.2426.
[210] Szpak N., Schutzhold R., Quantum simulator for the Schwinger effect with atoms in bichromatic optical
lattices, Phys. Rev. A 84 (2011), 050101(R), 4 pages, arXiv:1103.0541.
[211] Terazawa H., Various actions for pregeometry, Progr. Theoret. Phys. 86 (1991), 337–342.
[212] Terazawa H., Akama K., Dynamical subquark model of pregauge and pregeometric interactions, Phys.
Lett. B 96 (1980), 276-278.
[213] Terazawa H., Akama K., Chikashige Y., What are the gauge bosons made of?, Progr. Theoret. Phys. 56
(1976), 1935–1938.
[214] Terazawa H., Chikashige Y., Akama K., Unified model of the Nambu–Jona–Lasinio type for all elementary
particle forces, Phys. Rev. D 15 (1977), 480–487.
[215] Ueda M., Kawaguchi Y., Spinor Bose–Einstein condensates, arXiv:1001.2072.
[216] Unruh W.G., Experimental black-hole evaporation?, Phys. Rev. Lett. 46 (1981), 1351–1353.
[217] Unruh W.G., Sonic analogue of black holes and the effects of high frequencies on black hole evaporation,
Phys. Rev. D 51 (1995), 2827–2838.
Emergent Models for Gravity: an Overview of Microscopic Models
45
[218] Unruh W.G., Schützhold R., Universality of the Hawking effect, Phys. Rev. D 71 (2005), 024028, 11 pages,
gr-qc/0408009.
[219] Vacaru S., Stavrinos P., Gaburov E., Gonţa D., Clifford and Riemann–Finsler structures in geometric
mechanics and gravity, DGDS. Differential Geometry – Dynamical Systems. Monographs, Vol. 7, Geometry
Balkan Press, Bucharest, 2006, gr-qc/0508023.
[220] Verlinde E., On the origin of gravity and the laws of Newton, J. High Energy Phys. 2011 (2011), no. 4,
029, 27 pages, arXiv:1001.0785.
[221] Visser M., Acoustic propagation in fluids: an unexpected example of Lorentzian geometry, gr-qc/9311028.
[222] Visser M., Sakharov’s induced gravity: a modern perspective, Modern Phys. Lett. A 17 (2002), 977–991,
gr-qc/0204062.
[223] Visser M., Barceló C., Liberati S., Analogue models of and for gravity, Gen. Relativity Gravitation 34
(2002), 1719–1734, gr-qc/0111111.
[224] Volovik G.E., Cosmological constant and vacuum energy, Ann. Phys. 14 (2005), 165–176, gr-qc/0405012.
[225] Volovik G.E., Emergent physics: Fermi-point scenario, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys.
Eng. Sci. 366 (2008), 2935–2951, arXiv:0801.0724.
[226] Volovik G.E., Emergent physics on vacuum energy and cosmological constant, AIP Conf. Proc. 850 (2006),
26–33, cond-mat/0507454.
[227] Volovik G.E., Quantum phase transitions from topology in momentum space, in Quantum Analogues: from
Phase Transitions to Black Holes and Cosmology, Lecture Notes in Phys., Vol. 718, Springer, Berlin, 2007,
31–73, cond-mat/0601372.
[228] Volovik G.E., Superfluid analogies of cosmological phenomena, Phys. Rep. 351 (2001), 195–348,
gr-qc/0005091.
[229] Volovik G.E., The universe in a helium droplet, International Series of Monographs on Physics, Vol. 117,
The Clarendon Press, Oxford University Press, New York, 2003.
[230] Volovik G.E., Vacuum energy: quantum hydrodynamics versus quantum gravity, JETP Lett. 82 (2005),
319–324, gr-qc/0505104.
[231] Weinberg S., Photons and gravitons in S-matrix theory: derivation of charge conservation and equality of
gravitational and inertial mass, Phys. Rev. 135 (1964), B1049–B1056.
[232] Weinberg S., The cosmological constant problem, Rev. Modern Phys. 61 (1989), 1–23.
[233] Weinberg S., The quantum theory of fields. Vol. 1. Foundations, Cambridge University Press, Cambridge,
1995.
[234] Weinberg S., Witten E., Limits of massless particles, Phys. Lett. B 96 (1980), 59–62.
[235] Weinfurtner S., Jain P., Visser M., Gardiner C.W., Cosmological particle production in emergent rainbow
spacetimes, Classical Quantum Gravity 26 (2009), 065012, 49 pages, arXiv:0801.2673.
[236] Weinfurtner S., Liberati S., Visser M., Analogue space-time based on 2-component Bose–Einstein condensates, in Quantum Analogues: from Phase Transitions to Black Holes and Cosmology, Lecture Notes in
Phys., Vol. 718, Springer, Berlin, 2007, 115–163, gr-qc/0605121.
[237] Weinfurtner S., Tedford E.W., Penrice M.C.J., Unruh W.G., Lawrence G.A., Measurement of stimulated
Hawking emission in an analogue system, Phys. Rev. Lett. 106 (2011), 021302, 4 pages, arXiv:1008.1911.
[238] Weinfurtner S., White A., Visser M., Trans-Planckian physics and signature change events in Bose gas
hydrodynamics, Phys. Rev. D 76 (2007), 124008, 19 pages, gr-qc/0703117.
[239] Wen X.G., Quantum field theory of many-body systems: from the origin of sound to an origin of light and
electrons, Oxford University Press, Oxford, 2004.
[240] Wen X.G., Quantum order from string-net condensations and the origin of light and massless fermions,
Phys. Rev. D 68 (2003), 065003, 25 pages, hep-th/0302201.
[241] Wetterich C., Gravity from spinors, Phys. Rev. D 70 (2004), 105004, 21 pages, hep-th/0307145.
[242] Wetterich C., Lattice spinor gravity, Phys. Lett. B 704 (2011), 612–619, arXiv:1108.1313.
[243] Wetterich C., Spontaneous symmetry breaking origin for the difference between time and space, Phys. Rev.
Lett. 94 (2005), 011602, 4 pages.
[244] Yoshimoto S., Spinor pregeometry at finite temperature, Progr. Theoret. Phys. 78 (1987), 435–439.
[245] Zee A., Broken-symmetric theory of gravity, Phys. Rev. Lett. 42 (1979), 417–421.
[246] Zee A., Spontaneously generated gravity, Phys. Rev. D 23 (1981), 858–866.